Add Vectors in Different Ways - Precalculus

Two vectors are drawn tail-to-tail from the origin: $\mathbf{u}=\langle 4,1\rangle$ and $\mathbf{v}=\langle -1,3\rangle$ . Using the parallelogram rule, which vector represents $\mathbf{u}+\mathbf{v}$ (in component form)?
Answer: $\langle 3,4\rangle$ 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, 1⟩ and v = ⟨-1, 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 + (-1), 1 + 3⟩ = ⟨3, 4⟩. Choice B is correct because it properly applies the formula with specific values. Choice C incorrectly describes the parallelogram diagonal, using the wrong diagonal or wrong starting point, leading to ⟨-4, 3⟩. 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.
For vectors $\mathbf{u}=\langle 3,0\rangle$ and $\mathbf{v}=\langle 0,4\rangle$ , what is the magnitude of $\mathbf{u}+\mathbf{v}$ ? Give your answer in simplest form.
Answer: $5$ This question tests understanding of vector addition using component-wise addition and magnitude calculation. 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, 0⟩ and v = ⟨0, 4⟩, we compute u + v = ⟨3 + 0, 0 + 4⟩ = ⟨3, 4⟩. Computing |u + v| requires first finding the components of the sum ⟨3, 4⟩, then calculating √(3² + 4²) = √(9 + 16) = √25 = 5. Choice B is correct because it shows the proper magnitude calculation of the resultant vector. 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, which in this case are perpendicular. 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.
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)?
Answer: $\langle 2,6\rangle$ 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.
A boat's velocity relative to the water is $\langle 5,0\rangle$ m/s and the current's velocity is $\langle 0,2\rangle$ m/s. Using component-wise addition, what is the resultant velocity vector $\langle 5,0\rangle+\langle 0,2\rangle$ ?
Answer: $\langle 5,2\rangle$ 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 ⟨5, 0⟩ and ⟨0, 2⟩, we compute the sum = ⟨5 + 0, 0 + 2⟩ = ⟨5, 2⟩. Choice A is correct because it shows the component-wise calculation with specific values. Choice D adds only one component correctly but makes an arithmetic or sign error in the other component, leading to ⟨10, 0⟩ as if doubling the horizontal. 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.
For vectors $\mathbf{u}=\langle 3,0\rangle$ and $\mathbf{v}=\langle 0,4\rangle$ , what is the magnitude $\|\mathbf{u}+\mathbf{v}\|$ ? Give your answer in simplest form.
Answer: $5$ 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 ⟨3, 4⟩, then calculating √(3² + 4²) = 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.
A parallelogram has vertices at $O(0,0)$ , $A(5,2)$ , $B(7,6)$ , and $C(2,4)$ . If $\vec{OA}$ and $\vec{OC}$ represent two vectors being added using the parallelogram rule, which point represents the sum of these vectors?
Answer: Point $B(7,6)$ , which is the fourth vertex of the parallelogram When you encounter vector addition with parallelograms, remember that the parallelogram rule states that if two vectors form adjacent sides of a parallelogram starting from the same point, their sum is represented by the diagonal vector from that starting point to the opposite vertex. Here, vectors $\vec{OA} = (5,2)$ and $\vec{OC} = (2,4)$ both start at origin $O(0,0)$ . To find their sum using the parallelogram rule, you need to identify which point represents the end of the diagonal that starts at $O$ . We can verify this algebraically: $\vec{OA} + \vec{OC} = (5,2) + (2,4) = (7,6)$ , which corresponds exactly to point $B(7,6)$ . This makes perfect sense because in parallelogram $OACB$ , the diagonal from $O$ to $B$ represents the vector sum. Choice A incorrectly identifies the midpoint of diagonal $OB$ , but vector addition gives you the endpoint of the resultant vector, not its midpoint. Choice B points to the center of the parallelogram, which has no special meaning in vector addition. Choice C suggests the midpoint of diagonal $AC$ , but this diagonal doesn't start from the origin where our vectors begin, making it irrelevant to the sum $\vec{OA} + \vec{OC}$ . The answer is D: Point $B(7,6)$ . Key takeaway: In the parallelogram rule for vector addition, the sum always points from the common starting point to the opposite vertex of the parallelogram—never to midpoints or centers.
Points $A$ , $B$ , and $C$ form an end-to-end vector path where $\overrightarrow{AB}=\langle 2,1\rangle$ and $\overrightarrow{BC}=\langle 3,4\rangle$ . Using the vectors given, what is $\overrightarrow{AC}$ (in component form)?
Answer: $\langle 5,5\rangle$ 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 vector AB = ⟨2, 1⟩ from point A to point B, we place vector BC = ⟨3, 4⟩ so its tail is at B, terminating at point C. The sum AB + BC gives us AC, which geometrically represents the combined displacement from A to C. For vectors AB = ⟨2, 1⟩ and BC = ⟨3, 4⟩, we compute AC = ⟨2 + 3, 1 + 4⟩ = ⟨5, 5⟩. Choice B is correct because it shows the component-wise calculation of the path from A to C. Choice A subtracts the vectors instead of adding them, calculating ⟨2 - 3, 1 - 4⟩ = ⟨-1, -3⟩, which would represent going backwards along the path. 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.
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 $\langle x,y\rangle$ ?
Answer: $\langle 5,3\rangle$ 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.
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?
Answer: $8$ m, since the horizontal components cancel and only vertical displacement remains 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.
Based on the description, vector $\mathbf{u}$ has magnitude $4$ and direction due north, and vector $\mathbf{v}$ has magnitude $3$ and direction due east. What is the magnitude of $\mathbf{u}+\mathbf{v}$ ?
Answer: $5$ 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 ⟨3, 4⟩, then calculating √(3² + 4²) = 5, which is less than |u| + |v| = 7. Choice A 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.
For vectors $\mathbf{u}=\langle 3,0\rangle$ and $\mathbf{v}=\langle 0,4\rangle$ , what is the magnitude $\lVert \mathbf{u}+\mathbf{v} \rVert$ ? Give your answer in simplest form.
Answer: $5$ 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.
Refer to the coordinate plane. Vectors $\vec{AB}$ and $\vec{AC}$ are shown with $A$ at the origin. When these vectors are added using the end-to-end method, what is the correct sequence of points that traces the vector addition?
Answer: Start at $A$ , move to $B$ , then move parallel to $\vec{AC}$ to reach point $D$ In the end-to-end method for adding $\vec{AB} + \vec{AC}$ , we place the tail of the second vector at the head of the first. Starting at $A$ , we follow $\vec{AB}$ to reach $B$ , then from $B$ we move in the direction parallel to $\vec{AC}$ to reach the final point $D$ . The resultant $\vec{AD}$ represents the sum. Choice B describes adding in the opposite order. Choice C incorrectly uses midpoints. Choice D starts at the wrong point and doesn't represent proper vector addition.
Vectors $\mathbf{u}$ and $\mathbf{v}$ are added using the parallelogram rule (tails at the same initial point). Which statement correctly describes the sum $\mathbf{u}+\mathbf{v}$ ?
Answer: It is the diagonal of the parallelogram starting at the common initial point. 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 and v 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. Choice B is correct because it correctly describes the geometric construction of vector addition using the parallelogram rule. Choice C 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. 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.
Vector $\overrightarrow{AB}$ has components $\langle 2,1\rangle$ and vector $\overrightarrow{BC}$ has components $\langle 3,4\rangle$ . Using the end-to-end method (head-to-tail), what is the resultant vector $\overrightarrow{AC}$ in component form?
Answer: $\langle 5,5\rangle$ 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 vector AB = ⟨2, 1⟩ from point A to point B, we place vector BC = ⟨3, 4⟩ so its tail is at B, terminating at point C; the sum AB + BC is then the vector from A to C, which is ⟨2 + 3, 1 + 4⟩ = ⟨5, 5⟩. Choice C is correct because it correctly describes the geometric construction in component form. Choice B adds only one component correctly but makes an arithmetic or sign error in the other component, leading to ⟨6, 5⟩. 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.
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?
Answer: $10.8$ units, obtained by component-wise addition then magnitude calculation 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.
For vectors $\mathbf{u}=\langle 1,-2\rangle$ , $\mathbf{v}=\langle 3,5\rangle$ , and $\mathbf{w}=\langle -4,1\rangle$ , what is $\mathbf{u}+\mathbf{v}+\mathbf{w}$ in component form?
Answer: $\langle 0,4\rangle$ 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 = ⟨1, -2⟩, v = ⟨3, 5⟩, and w = ⟨-4, 1⟩, we compute u + v + w = ⟨1 + 3 + (-4), -2 + 5 + 1⟩ = ⟨0, 4⟩. Choice A is correct because it shows the component-wise calculation. Choice D adds only one component correctly but makes an arithmetic or sign error in the other component. 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.
A hiker walks $5$ units east and then $-2$ units east (i.e., $2$ units west). Treat these as vectors on the $x$ -axis: $\mathbf{u}=\langle 5,0\rangle$ and $\mathbf{v}=\langle -2,0\rangle$ . What are the components of $\mathbf{u}+\mathbf{v}$ ?
Answer: $\langle 3,0\rangle$ 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 = ⟨5, 0⟩ and v = ⟨-2, 0⟩, we compute u + v = ⟨5 + (-2), 0 + 0⟩ = ⟨3, 0⟩. Choice A is correct because it shows the component-wise calculation with specific values. Choice C subtracts the vectors instead of adding them, calculating ⟨5 - (-2), 0 - 0⟩ but mistakenly as ⟨-3, 0⟩, which is not the sum. 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. Remember that adding a negative component is like moving in the opposite direction, but the method remains the same.
For vectors $\mathbf{u}=\langle 4,1\rangle$ and $\mathbf{v}=\langle -2,3\rangle$ , what is $\mathbf{u}+\mathbf{v}$ (in component form)?
Answer: $\langle 2,4\rangle$ 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.
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 $\mathbf{v}$ ?
Answer: $\|\mathbf{u}+\mathbf{v}\|\le\|\mathbf{u}\|+\|\mathbf{v}\|$ always 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.
Three vectors are given: $\mathbf{u}=\langle 1,2\rangle$ , $\mathbf{v}=\langle 3,-1\rangle$ , and $\mathbf{w}=\langle -2,4\rangle$ . Using component-wise addition, what is $\mathbf{u}+\mathbf{v}+\mathbf{w}$ ?
Answer: $\langle 2,5\rangle$ 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 = ⟨1, 2⟩, v = ⟨3, -1⟩, and w = ⟨-2, 4⟩, we compute u + v + w = ⟨1 + 3 + (-2), 2 + (-1) + 4⟩ = ⟨2, 5⟩. Choice A is correct because it shows the component-wise calculation for all three vectors. Choice B reverses the component order, computing something like ⟨6, 1⟩, but component order matters (first is horizontal, second is vertical). 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.

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