Precalculus Quiz: Add Vectors In Different Ways

20 questions · exam conditions

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Add Vectors In Different WaysQuestion 1 of 20

Based on the triangle inequality for vector addition, how does $\|\mathbf{u}+\mathbf{v}\|$ compare to $\|\mathbf{u}\|+\|\mathbf{v}\|$ for any vectors $\mathbf{u}$ and $\mathbf{v}$ ?

\|\mathbf{u}+\mathbf{v}\|>\|\mathbf{u}\|+\|\mathbf{v}\|

always

\|\mathbf{u}+\mathbf{v}\|=\|\mathbf{u}\|+\|\mathbf{v}\|

always

\|\mathbf{u}+\mathbf{v}\|\le\|\mathbf{u}\|+\|\mathbf{v}\|

always

\|\mathbf{u}+\mathbf{v}\|\ge\|\mathbf{u}\|+\|\mathbf{v}\|

always

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Precalculus Quiz

Precalculus Quiz: Add Vectors In Different Ways

Practice Add Vectors In Different Ways in Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

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This quiz focuses on Add Vectors In Different Ways, giving you a quick way to practice the rules, question types, and explanations that matter most for Precalculus.

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Question 1

Based on the triangle inequality for vector addition, how does $\|\mathbf{u}+\mathbf{v}\|$ compare to $\|\mathbf{u}\|+\|\mathbf{v}\|$ for any vectors $\mathbf{u}$ and $\mathbf{v}$ ?

$\|\mathbf{u}+\mathbf{v}\|>\|\mathbf{u}\|+\|\mathbf{v}\|$ always
$\|\mathbf{u}+\mathbf{v}\|=\|\mathbf{u}\|+\|\mathbf{v}\|$ always
$\|\mathbf{u}+\mathbf{v}\|\le\|\mathbf{u}\|+\|\mathbf{v}\|$ always (correct answer)
$\|\mathbf{u}+\mathbf{v}\|\ge\|\mathbf{u}\|+\|\mathbf{v}\|$ always

Explanation: This question tests understanding of vector addition using multiple methods. An important property of vector addition is that the magnitude of the sum is typically not equal to the sum of the magnitudes: |u + v| ≤ |u| + |v|, with equality only when the vectors point in exactly the same direction. Computing |u + v| requires first finding the components of the sum, then calculating the magnitude, which is always less than or equal to |u| + |v|. Choice C is correct because it properly applies the formula with specific values. Choice B incorrectly assumes that |u + v| = |u| + |v|, but vector magnitude addition doesn't work this way—the magnitude of a sum depends on the angle between the vectors. Remember that |u + v| ≠ |u| + |v| in general: the magnitude of a vector sum depends on both the magnitudes and the angle between the vectors, reaching its maximum (|u| + |v|) only when vectors point the same direction. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.

Question 2

Vectors $\mathbf{u}=\langle 2,3\rangle$ and $\mathbf{v}=\langle -2,-3\rangle$ are additive inverses. What is $\mathbf{u}+\mathbf{v}$ ?

$\langle 4,6\rangle$
$\langle 0,0\rangle$ (correct answer)
$\langle -4,-6\rangle$
$\langle 0,1\rangle$

Explanation: This question tests understanding of vector addition using the component-wise method with additive inverses. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨2, 3⟩ and v = ⟨-2, -3⟩, we compute u + v = ⟨2 + (-2), 3 + (-3)⟩ = ⟨0, 0⟩. Choice B is correct because additive inverse vectors have opposite components that cancel when added: 2 + (-2) = 0 and 3 + (-3) = 0, resulting in the zero vector. Choice A adds the absolute values instead of the signed values, computing ⟨|2| + |-2|, |3| + |-3|⟩ = ⟨4, 6⟩, which ignores the negative signs. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, and remember that additive inverses always sum to the zero vector ⟨0, 0⟩.

Question 3

Vectors $\mathbf{u}=\langle 1,-2\rangle$ , $\mathbf{v}=\langle 4,3\rangle$ , and $\mathbf{w}=\langle -2,1\rangle$ are given. Using component-wise addition, what is $\mathbf{u}+\mathbf{v}+\mathbf{w}$ ?

$\langle 3,2\rangle$ (correct answer)
$\langle 7,2\rangle$
$\langle 3,0\rangle$
$\langle -5,-4\rangle$

Explanation: This question tests understanding of vector addition using the component-wise method with three vectors. Vectors are added component-wise: for multiple vectors, we add all corresponding components together separately. For vectors u = ⟨1, -2⟩, v = ⟨4, 3⟩, and w = ⟨-2, 1⟩, we compute u + v + w = ⟨1 + 4 + (-2), -2 + 3 + 1⟩ = ⟨3, 2⟩. Choice A is correct because it shows the component-wise calculation: the x-components give 1 + 4 - 2 = 3, and the y-components give -2 + 3 + 1 = 2. Choice B adds only the first two vectors correctly but makes an error with the third, computing ⟨1 + 4 + 2, -2 + 3 + 1⟩ = ⟨7, 2⟩, likely misreading the sign of w's first component. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—for multiple vectors, component-wise addition is often the most efficient approach.

Question 4

Two vectors are drawn from the origin: $\mathbf{u}=\langle 4,0\rangle$ and $\mathbf{v}=\langle 0,3\rangle$ . Using the parallelogram rule, what is $\mathbf{u}+\mathbf{v}$ in component form?

$\langle 4,3\rangle$ (correct answer)
$\langle 1,12\rangle$
$\langle 4,-3\rangle$
$\langle 3,4\rangle$

Explanation: This question tests understanding of vector addition using the parallelogram rule. The parallelogram rule for adding vectors involves placing both vectors tail-to-tail at a common initial point, completing a parallelogram using the vectors as adjacent sides, and taking the diagonal from the initial point as the sum. When both vectors u = ⟨4, 0⟩ and v = ⟨0, 3⟩ start from the same point, they form two adjacent sides of a parallelogram; the diagonal of this parallelogram starting from the common initial point represents u + v = ⟨4, 3⟩. Choice A is correct because it properly applies the formula with specific values in component form. Choice C subtracts the vectors instead of adding them, calculating ⟨4 - 0, 0 - 3⟩, which gives a different vector. The parallelogram rule shows why vector addition is commutative (u + v = v + u): both diagonals of the parallelogram represent the same sum, just constructed in different orders. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.

Question 5

For vectors $\mathbf{u}=\langle 4,1\rangle$ and $\mathbf{v}=\langle -2,3\rangle$ , what is $\mathbf{u}+\mathbf{v}$ (in component form)?

$\langle 2,4\rangle$ (correct answer)
$\langle -8,3\rangle$
$\langle 2,-2\rangle$
$\langle -1,4\rangle$

Explanation: This question tests understanding of vector addition using the component-wise method. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨4, 1⟩ and v = ⟨-2, 3⟩, we compute u + v = ⟨4 + (-2), 1 + 3⟩ = ⟨2, 4⟩. Choice A is correct because it shows the component-wise calculation: the x-components give 4 + (-2) = 2, and the y-components give 1 + 3 = 4. Choice B multiplies corresponding components together instead of adding them, computing ⟨4(-2), 1(3)⟩ = ⟨-8, 3⟩, which is not a standard vector operation. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, like combining horizontal and vertical motions.

Question 6

A hiker walks $5$ units east and then $3$ units north. Treat these as vectors added head-to-tail. What are the components of the resultant displacement vector in the form $\langle x,y\rangle$ ?

$\langle 8,0\rangle$
$\langle 5,3\rangle$ (correct answer)
$\langle 2,15\rangle$
$\langle 3,5\rangle$

Explanation: This question tests understanding of vector addition using the end-to-end method. The end-to-end (or head-to-tail) method of vector addition involves placing the tail of the second vector at the head of the first vector; the resultant vector then goes from the tail of the first vector to the head of the second vector. Starting with the vector 5 units east (⟨5, 0⟩) from point A to point B, we place the vector 3 units north (⟨0, 3⟩) so its tail is at B, terminating at point C; the sum is then the vector from A to C, which is ⟨5, 3⟩. Choice B is correct because it correctly describes the geometric construction in component form. Choice A incorrectly assumes that |u + v| = |u| + |v|, computing ⟨8, 0⟩ as if both displacements were in the same direction. For end-to-end addition, always place the tail (starting point) of the second vector at the head (endpoint) of the first vector, then draw the resultant from the original starting point to the final ending point. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.

Question 7

For vectors $\mathbf{u}=\langle 2,0\rangle$ and $\mathbf{v}=\langle 0,2\rangle$ , how does $\lVert \mathbf{u}+\mathbf{v} \rVert$ compare to $\lVert \mathbf{u} \rVert + \lVert \mathbf{v} \rVert$ ?

$\lVert \mathbf{u}+\mathbf{v} \rVert > \lVert \mathbf{u} \rVert + \lVert \mathbf{v} \rVert$
$\lVert \mathbf{u}+\mathbf{v} \rVert = \lVert \mathbf{u} \rVert + \lVert \mathbf{v} \rVert$
$\lVert \mathbf{u}+\mathbf{v} \rVert < \lVert \mathbf{u} \rVert + \lVert \mathbf{v} \rVert$ (correct answer)
$\lVert \mathbf{u}+\mathbf{v} \rVert$ cannot be compared to $\lVert \mathbf{u} \rVert + \lVert \mathbf{v} \rVert$ without a calculator

Explanation: This question tests understanding of the relationship between vector magnitude and vector addition. An important property of vector addition is that the magnitude of the sum is typically not equal to the sum of the magnitudes: |u + v| ≤ |u| + |v|, with equality only when the vectors point in exactly the same direction. For vectors u = ⟨2, 0⟩ and v = ⟨0, 2⟩, we first compute u + v = ⟨2 + 0, 0 + 2⟩ = ⟨2, 2⟩. Computing |u + v| requires calculating √(2² + 2²) = √8 = 2√2 ≈ 2.83, which is less than |u| + |v| = 2 + 2 = 4. Choice C is correct because |u + v| = 2√2 < 4 = |u| + |v|, since the vectors are perpendicular (one points east, one points north) rather than in the same direction. Choice B incorrectly assumes that |u + v| = |u| + |v|, but vector magnitude addition doesn't work this way—the magnitude of a sum depends on the angle between the vectors. Remember that |u + v| ≠ |u| + |v| in general: the magnitude of a vector sum depends on both the magnitudes and the angle between the vectors, reaching its maximum (|u| + |v|) only when vectors point the same direction.

Question 8

For vectors $\mathbf{u}=\langle -6,2\rangle$ and $\mathbf{v}=\langle 1,-5\rangle$ , what is $\mathbf{u}+\mathbf{v}$ (in component form)?

$\langle -5,-3\rangle$ (correct answer)
$\langle -7,7\rangle$
$\langle -6,-10\rangle$
$\langle -6,2\rangle$

Explanation: This question tests understanding of vector addition using the component-wise method with negative components. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨-6, 2⟩ and v = ⟨1, -5⟩, we compute u + v = ⟨-6 + 1, 2 + (-5)⟩ = ⟨-5, -3⟩. Choice A is correct because it shows the component-wise calculation: the x-components give -6 + 1 = -5, and the y-components give 2 + (-5) = -3. Choice B subtracts the vectors instead of adding them, calculating ⟨-6 - 1, 2 - (-5)⟩ = ⟨-7, 7⟩, which gives u - v rather than u + v. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, being careful with negative signs.

Question 9

Vector $\mathbf{u}=\langle 3,4\rangle$ and vector $\mathbf{v}=\langle -1,2\rangle$ . Using component-wise addition, what are the components of the sum $\mathbf{u}+\mathbf{v}$ ?

$\langle 4,2\rangle$
$\langle 2,6\rangle$ (correct answer)
$\langle -3,8\rangle$
$\langle -3,6\rangle$

Explanation: This question tests understanding of vector addition using the component-wise method. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨3, 4⟩ and v = ⟨-1, 2⟩, we compute u + v = ⟨3 + (-1), 4 + 2⟩ = ⟨2, 6⟩. Choice B is correct because it shows the component-wise calculation: the x-components give 3 + (-1) = 2, and the y-components give 4 + 2 = 6. Choice A subtracts the vectors instead of adding them, calculating ⟨3 - (-1), 4 - 2⟩ = ⟨4, 2⟩, which gives u - v rather than u + v. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, like combining horizontal and vertical motions.

Question 10

Let $\mathbf{u}=\langle 3,0\rangle$ and $\mathbf{v}=\langle 0,4\rangle$ . How does $\lvert \mathbf{u}+\mathbf{v} \rvert$ compare to $\lvert \mathbf{u} \rvert + \lvert \mathbf{v} \rvert$ ?

$\lvert \mathbf{u}+\mathbf{v} \rvert = \lvert \mathbf{u} \rvert + \lvert \mathbf{v} \rvert$
$\lvert \mathbf{u}+\mathbf{v} \rvert > \lvert \mathbf{u} \rvert + \lvert \mathbf{v} \rvert$
$\lvert \mathbf{u}+\mathbf{v} \rvert < \lvert \mathbf{u} \rvert + \lvert \mathbf{v} \rvert$ (correct answer)
$\lvert \mathbf{u}+\mathbf{v} \rvert$ cannot be compared to $\lvert \mathbf{u} \rvert + \lvert \mathbf{v} \rvert$ without more information

|u + v| \leq |u| + |v|

, with equality only when the vectors point in exactly the same direction. Computing

|u + v|

requires first finding the components of the sum

\langle 3, 4 \rangle

, then calculating

\sqrt{(3^2 + 4^2)} = \sqrt{(9 + 16)} = \sqrt{25} = 5

, which is less than

|u| + |v| = 3 + 4 = 7

. Choice C is correct because it properly applies the formula with specific values, showing the inequality holds. Choice A incorrectly assumes that

|u + v| = |u| + |v|

, but vector magnitude addition doesn't work this way—the magnitude of a sum depends on the angle between the vectors. Remember that

|u + v| \neq |u| + |v|

in general: the magnitude of a vector sum depends on both the magnitudes and the angle between the vectors, reaching its maximum (

|u| + |v|

) only when vectors point the same direction. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.

Question 11

For vectors $\mathbf{u}=\langle 3,4\rangle$ and $\mathbf{v}=\langle -1,2\rangle$ , using component-wise addition, what is $\mathbf{u}+\mathbf{v}$ in component form?

$\langle 2,2\rangle$
$\langle -3,8\rangle$
$\langle 2,6\rangle$ (correct answer)
$\langle -4,8\rangle$

Explanation: This question tests understanding of vector addition using the component-wise method. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨3, 4⟩ and v = ⟨-1, 2⟩, we compute u + v = ⟨3 + (-1), 4 + 2⟩ = ⟨2, 6⟩. Choice C is correct because it shows the component-wise calculation with specific values. Choice B subtracts the vectors instead of adding them, calculating ⟨3 - (-1), 4 - 2⟩ but with sign errors leading to ⟨-3, 8⟩, which gives something like a mistaken subtraction. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, like combining horizontal and vertical motions. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.

Question 12

For vectors $\mathbf{u}=\langle 6,1\rangle$ and $\mathbf{v}=\langle -2,5\rangle$ , what are the components of the sum $\mathbf{u}+\mathbf{v}$ ?

$\langle 4,6\rangle$ (correct answer)
$\langle 8,-4\rangle$
$\langle -12,5\rangle$
$\langle 6,10\rangle$

Explanation: This question tests understanding of vector addition using the component-wise method. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨6, 1⟩ and v = ⟨-2, 5⟩, we compute u + v = ⟨6 + (-2), 1 + 5⟩ = ⟨4, 6⟩. Choice A is correct because it shows the component-wise calculation with accurate addition. Choice B adds only one component correctly but makes an arithmetic or sign error in the other component, possibly doubling or negating incorrectly. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, like combining horizontal and vertical motions. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.

Question 13

For vectors $\mathbf{u}=\langle 3,4\rangle$ and $\mathbf{v}=\langle -1,2\rangle$ , using component-wise addition, what are the components of $\mathbf{u}+\mathbf{v}$ ?

$\langle 2,2\rangle$
$\langle -3,8\rangle$
$\langle 2,6\rangle$ (correct answer)
$\langle -3,2\rangle$

Explanation: This question tests understanding of vector addition using the component-wise method. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨3, 4⟩ and v = ⟨-1, 2⟩, we compute u + v = ⟨3 + (-1), 4 + 2⟩ = ⟨2, 6⟩. Choice C is correct because it shows the component-wise calculation: the x-components give 3 + (-1) = 2, and the y-components give 4 + 2 = 6. Choice D incorrectly subtracts the vectors instead of adding them, calculating ⟨3 - (-1), 4 - 2⟩ = ⟨4, 2⟩, which gives u - v rather than u + v. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, like combining horizontal and vertical motions.

Question 14

Vectors $\vec{u} = \langle 3, -2 \rangle$ and $\vec{v} = \langle -1, 4 \rangle$ are added using the parallelogram rule. If the parallelogram has vertices at the origin, point $P$ , point $Q$ , and point $R$ , where $\vec{OP} = \vec{u}$ and $\vec{OQ} = \vec{v}$ , what are the coordinates of vertex $R$ ?

$R = (2, 2)$ , representing the sum $\vec{u} + \vec{v}$ (correct answer)
$R = (4, -6)$ , representing the difference $\vec{u} - \vec{v}$
$R = (-4, 6)$ , representing the difference $\vec{v} - \vec{u}$
$R = (3, 4)$ , representing the diagonal of the parallelogram

Explanation: In the parallelogram rule, if we place vectors

\vec{u}

and

\vec{v}

at the origin with endpoints at

P(3, -2)

and

Q(-1, 4)

, then the fourth vertex

R

completes the parallelogram. Since

\vec{OR} = \vec{OP} + \vec{OQ} = \vec{u} + \vec{v}

, we have

R = (3 + (-1), -2 + 4) = (2, 2)

. Choice B incorrectly subtracts the vectors. Choice C gives the wrong order of subtraction. Choice D incorrectly uses individual components rather than vector addition.

Question 15

Two forces $\vec{F_1} = \langle 8, 6 \rangle$ and $\vec{F_2} = \langle -3, 4 \rangle$ act on an object. The magnitude of the resultant force is closest to which value?

$15$ units, obtained by adding the individual magnitudes
$5$ units, obtained by subtracting the smaller from larger magnitude
$10.8$ units, obtained by component-wise addition then magnitude calculation (correct answer)
$7.2$ units, obtained by averaging the two individual magnitudes

Explanation: The resultant force is

\vec{F_1} + \vec{F_2} = \langle 8-3, 6+4 \rangle = \langle 5, 10 \rangle

. Its magnitude is

\sqrt{5^2 + 10^2} = \sqrt{125} \approx 10.8

. Choice A incorrectly adds magnitudes:

|\vec{F_1}| + |\vec{F_2}| = 10 + 5 = 15

. Choice B incorrectly subtracts magnitudes. Choice D incorrectly averages the magnitudes. This demonstrates that the magnitude of a sum is typically not the sum of magnitudes.

Question 16

Two vectors $\vec{p}$ and $\vec{q}$ have magnitudes $|\vec{p}| = 7$ and $|\vec{q}| = 3$ . The angle between them is $120°$ . If a student incorrectly calculates the magnitude of $\vec{p} + \vec{q}$ as $10$ , what conceptual error did they make?

They computed the magnitude of the difference $\vec{p} - \vec{q}$ instead of the sum
They used the wrong formula, applying $|\vec{p} + \vec{q}|^2 = |\vec{p}|^2 - |\vec{q}|^2 + 2|\vec{p}||\vec{q}|\cos(120°)$
They calculated the magnitude using $120°$ instead of its supplement $60°$
They added the magnitudes directly, ignoring the angle between the vectors entirely (correct answer)

Explanation: When finding the magnitude of a vector sum, you need to account for both the magnitudes of the individual vectors AND the angle between them. This isn't simple arithmetic addition. The correct approach uses the law of cosines for vectors:

|\vec{p} + \vec{q}|^2 = |\vec{p}|^2 + |\vec{q}|^2 + 2|\vec{p}||\vec{q}|\cos(\theta)

. With

|\vec{p}| = 7

|\vec{q}| = 3

, and

\theta = 120°

|\vec{p} + \vec{q}|^2 = 7^2 + 3^2 + 2(7)(3)\cos(120°) = 49 + 9 + 42(-\frac{1}{2}) = 58 - 21 = 37

|\vec{p} + \vec{q}| = \sqrt{37} \approx 6.08

, not 10. Looking at the incorrect answer of 10: this equals

7 + 3

, which reveals the error. Choice D correctly identifies that the student simply added the magnitudes directly (

|\vec{p}| + |\vec{q}| = 7 + 3 = 10

), completely ignoring the angle between the vectors. Choice A is wrong because

|\vec{p} - \vec{q}| = \sqrt{37}

as well (the cosine term has the same result). Choice B describes an impossible formula with subtraction instead of addition. Choice C suggests using

60°

instead of

120°

, but this would give

|\vec{p} + \vec{q}| = \sqrt{79} \approx 8.89

, not 10. Remember: Vector addition is NOT scalar addition. When vectors aren't parallel, their magnitudes don't simply add. Always use the vector addition formula that incorporates the angle between them.

Question 17

Based on the description, $\mathbf{u}$ is $3$ units east and $\mathbf{v}$ is $4$ units north (so $\mathbf{u}=\langle 3,0\rangle$ and $\mathbf{v}=\langle 0,4\rangle$ ). How does $\lVert \mathbf{u}+\mathbf{v} \rVert$ compare to $\lVert \mathbf{u} \rVert+\lVert \mathbf{v} \rVert$ ?

$\lVert \mathbf{u}+\mathbf{v} \rVert = \lVert \mathbf{u} \rVert+\lVert \mathbf{v} \rVert$
$\lVert \mathbf{u}+\mathbf{v} \rVert > \lVert \mathbf{u} \rVert+\lVert \mathbf{v} \rVert$
$\lVert \mathbf{u}+\mathbf{v} \rVert < \lVert \mathbf{u} \rVert+\lVert \mathbf{v} \rVert$ (correct answer)
$\lVert \mathbf{u}+\mathbf{v} \rVert = \lVert \mathbf{u} \rVert-\lVert \mathbf{v} \rVert$

Explanation: This question tests understanding of vector addition using multiple methods, focusing on the magnitude relationship. An important property of vector addition is that the magnitude of the sum is typically not equal to the sum of the magnitudes: |u + v| ≤ |u| + |v|, with equality only when the vectors point in exactly the same direction. Computing |u + v| requires first finding the components of the sum ⟨3, 4⟩, then calculating √(3² + 4²) = 5, which is less than |u| + |v| = 3 + 4 = 7. Choice C is correct because it properly applies the formula with specific values. Choice A incorrectly assumes that |u + v| = |u| + |v|, but vector magnitude addition doesn't work this way—the magnitude of a sum depends on the angle between the vectors. Remember that |u + v| ≠ |u| + |v| in general: the magnitude of a vector sum depends on both the magnitudes and the angle between the vectors, reaching its maximum (|u| + |v|) only when vectors point the same direction. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.

Question 18

A drone moves according to three displacement vectors: $\vec{v_1} = \langle 6, 0 \rangle$ m, $\vec{v_2} = \langle 0, 8 \rangle$ m, and $\vec{v_3} = \langle -6, 0 \rangle$ m. Using the end-to-end method, what is the magnitude of the total displacement after all three moves?

$20$ m, obtained by adding all three individual displacement magnitudes together
$8$ m, since the horizontal components cancel and only vertical displacement remains (correct answer)
$10$ m, obtained by using the Pythagorean theorem on the original triangle
$0$ m, since the drone returns to its original horizontal position

Explanation: When working with vector displacement problems, you need to understand that the end-to-end method means adding vectors component-wise to find the resultant vector, then calculating its magnitude. Let's add the three displacement vectors component by component:

x-components: $6 + 0 + (-6) = 0$
y-components: $0 + 8 + 0 = 8$

The resultant displacement vector is

\langle 0, 8 \rangle

m. The magnitude of this vector is simply

\sqrt{0^2 + 8^2} = 8

m. Since the horizontal components completely cancel out (

+6

and

-6

), only the vertical displacement of

8

m remains. Choice A incorrectly adds the magnitudes of individual vectors (

6 + 8 + 6 = 20

) rather than using vector addition. You can't just add magnitudes when vectors point in different directions. Choice C mentions using the Pythagorean theorem on "the original triangle," but this doesn't make sense in the context of sequential displacements. While

\sqrt{6^2 + 8^2} = 10

would be correct for finding the hypotenuse of a right triangle with legs 6 and 8, it doesn't apply to this step-by-step motion. Choice D incorrectly focuses only on horizontal position, ignoring that the drone moved vertically and stayed there. Returning to the original horizontal position doesn't mean zero total displacement. Study tip: For vector problems, always work component-wise first. Add all x-components together, add all y-components together, then find the magnitude of the resulting vector. Don't add individual magnitudes unless all vectors point in the same direction.

Question 19

For vectors $\mathbf{u}=\langle 3,4\rangle$ and $\mathbf{v}=\langle -1,2\rangle$ , using component-wise addition, what is $\mathbf{u}+\mathbf{v}$ (in component form)?

$\langle 2,6\rangle$ (correct answer)
$\langle 4,2\rangle$
$\langle -3,8\rangle$
$\langle -3,2\rangle$

Explanation: This question tests understanding of vector addition using the component-wise method. Vectors are added component-wise: for u = ⟨u₁, u₂⟩ and v = ⟨v₁, v₂⟩, the sum u + v = ⟨u₁ + v₁, u₂ + v₂⟩, meaning we add the horizontal components together and the vertical components together separately. For vectors u = ⟨3, 4⟩ and v = ⟨-1, 2⟩, we compute u + v = ⟨3 + (-1), 4 + 2⟩ = ⟨2, 6⟩. Choice A is correct because it shows the component-wise calculation. Choice C multiplies corresponding components together instead of adding them, computing ⟨3*(-1), 4*2⟩ = ⟨-3, 8⟩, which is not a standard vector operation. Key to vector addition: when using component form, add x-components together and y-components together separately—treat each component independently, like combining horizontal and vertical motions. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.

Question 20

For vectors $\mathbf{u}=\langle 3,0\rangle$ and $\mathbf{v}=\langle 0,4\rangle$ , what is the magnitude $\lVert \mathbf{u}+\mathbf{v} \rVert$ ?

$7$
$5$ (correct answer)
$\sqrt{7}$
$\sqrt{25}$

Explanation: This question tests understanding of vector addition using multiple methods, focusing on the magnitude of the sum. An important property of vector addition is that the magnitude of the sum is typically not equal to the sum of the magnitudes: |u + v| ≤ |u| + |v|, with equality only when the vectors point in exactly the same direction. Computing |u + v| requires first finding the components of the sum ⟨3, 4⟩, then calculating √(3² + 4²) = √(9 + 16) = √25 = 5, which is less than |u| + |v| = 3 + 4 = 7. Choice B is correct because it properly applies the formula with specific values. Choice A incorrectly assumes that |u + v| = |u| + |v|, but vector magnitude addition doesn't work this way—the magnitude of a sum depends on the angle between the vectors. Remember that |u + v| ≠ |u| + |v| in general: the magnitude of a vector sum depends on both the magnitudes and the angle between the vectors, reaching its maximum (|u| + |v|) only when vectors point the same direction. All three methods (component-wise, end-to-end, parallelogram) give the same result because they're equivalent ways of representing the same mathematical operation—choose the method that best fits the context or presentation.