Vectors - AP Precalculus
Card 1 of 30
What operation is used to find a vector's component along another vector?
What operation is used to find a vector's component along another vector?
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Projection. Projects one vector onto the direction of another.
Projection. Projects one vector onto the direction of another.
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State the formula for the projection of $\textbf{u}$ onto $\textbf{v}$.
State the formula for the projection of $\textbf{u}$ onto $\textbf{v}$.
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$\text{proj}_{\textbf{v}} \textbf{u} = \frac{\textbf{u} \bullet \textbf{v}}{|\textbf{v}|^2} \textbf{v}$. Uses dot product and magnitude to find the component.
$\text{proj}_{\textbf{v}} \textbf{u} = \frac{\textbf{u} \bullet \textbf{v}}{|\textbf{v}|^2} \textbf{v}$. Uses dot product and magnitude to find the component.
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Identify the property: $\textbf{u} + \textbf{v} = \textbf{v} + \textbf{u}$.
Identify the property: $\textbf{u} + \textbf{v} = \textbf{v} + \textbf{u}$.
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Commutative property of vector addition. Vector addition is commutative like regular addition.
Commutative property of vector addition. Vector addition is commutative like regular addition.
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What is the magnitude of $\begin{bmatrix} 0 \\ 0 \\ 0 \matrix}$?
What is the magnitude of $\begin{bmatrix} 0 \\ 0 \\ 0 \matrix}$?
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Zero. The zero vector has zero magnitude by definition.
Zero. The zero vector has zero magnitude by definition.
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Find the scalar projection of $\begin{bmatrix} 3 \\ 4 \matrix}$ on $\begin{bmatrix} 4 \\ 3 \matrix}$.
Find the scalar projection of $\begin{bmatrix} 3 \\ 4 \matrix}$ on $\begin{bmatrix} 4 \\ 3 \matrix}$.
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$\frac{24}{5}$. Scalar projection: $\frac{\textbf{u} \cdot \textbf{v}}{|\textbf{v}|} = \frac{24}{5}$.
$\frac{24}{5}$. Scalar projection: $\frac{\textbf{u} \cdot \textbf{v}}{|\textbf{v}|} = \frac{24}{5}$.
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State the formula for the projection of $\textbf{u}$ onto $\textbf{v}$.
State the formula for the projection of $\textbf{u}$ onto $\textbf{v}$.
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$\text{proj}_{\textbf{v}} \textbf{u} = \frac{\textbf{u} \bullet \textbf{v}}{|\textbf{v}|^2} \textbf{v}$. Uses dot product and magnitude to find the component.
$\text{proj}_{\textbf{v}} \textbf{u} = \frac{\textbf{u} \bullet \textbf{v}}{|\textbf{v}|^2} \textbf{v}$. Uses dot product and magnitude to find the component.
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Which condition indicates that two vectors are parallel?
Which condition indicates that two vectors are parallel?
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One vector is a scalar multiple of the other. Parallel vectors point in the same or opposite directions.
One vector is a scalar multiple of the other. Parallel vectors point in the same or opposite directions.
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What is the magnitude of the zero vector?
What is the magnitude of the zero vector?
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Zero. The zero vector has no length by definition.
Zero. The zero vector has no length by definition.
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Find the vector sum of $\textbf{a} = \begin{bmatrix} 3 \\ -2 \matrix}$ and $\textbf{b} = \begin{bmatrix} -1 \\ 4 \matrix}$.
Find the vector sum of $\textbf{a} = \begin{bmatrix} 3 \\ -2 \matrix}$ and $\textbf{b} = \begin{bmatrix} -1 \\ 4 \matrix}$.
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$\begin{bmatrix} 2 \\ 2 \matrix}$. Add corresponding components: $(3-1, -2+4) = (2,2)$.
$\begin{bmatrix} 2 \\ 2 \matrix}$. Add corresponding components: $(3-1, -2+4) = (2,2)$.
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Find the dot product of $\begin{bmatrix} 1 \\ 3 \matrix}$ and $\begin{bmatrix} 4 \\ -2 \matrix}$.
Find the dot product of $\begin{bmatrix} 1 \\ 3 \matrix}$ and $\begin{bmatrix} 4 \\ -2 \matrix}$.
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$-2$. Calculate: $(1)(4) + (3)(-2) = 4 - 6 = -2$.
$-2$. Calculate: $(1)(4) + (3)(-2) = 4 - 6 = -2$.
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What is the direction of a vector if its components are all equal?
What is the direction of a vector if its components are all equal?
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The direction is along the line $y=x=z$. Equal components create a vector along the main diagonal.
The direction is along the line $y=x=z$. Equal components create a vector along the main diagonal.
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Determine if vectors $\begin{bmatrix} 2 \\ 3 \matrix}$ and $\begin{bmatrix} -4 \\ -6 \matrix}$ are parallel.
Determine if vectors $\begin{bmatrix} 2 \\ 3 \matrix}$ and $\begin{bmatrix} -4 \\ -6 \matrix}$ are parallel.
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Yes, they are parallel. The second vector is $-2$ times the first vector.
Yes, they are parallel. The second vector is $-2$ times the first vector.
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Find the magnitude of vector $\begin{bmatrix} 3 \\ 4 \matrix}$.
Find the magnitude of vector $\begin{bmatrix} 3 \\ 4 \matrix}$.
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$5$. Use the formula $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.
$5$. Use the formula $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.
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What is the result of multiplying a vector by a scalar $k$?
What is the result of multiplying a vector by a scalar $k$?
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A vector in the same direction if $k > 0$, opposite if $k < 0$. Scales magnitude and may reverse direction based on sign.
A vector in the same direction if $k > 0$, opposite if $k < 0$. Scales magnitude and may reverse direction based on sign.
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Find the unit vector of $\textbf{v} = \begin{bmatrix} 5 \\ 12 \matrix}$.
Find the unit vector of $\textbf{v} = \begin{bmatrix} 5 \\ 12 \matrix}$.
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$\begin{bmatrix} \frac{5}{13} \\ \frac{12}{13} \matrix}$. Divide by magnitude $13$ to get unit vector.
$\begin{bmatrix} \frac{5}{13} \\ \frac{12}{13} \matrix}$. Divide by magnitude $13$ to get unit vector.
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What is meant by the term 'orthogonal vectors'?
What is meant by the term 'orthogonal vectors'?
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Vectors with a dot product of zero. Perpendicular vectors meet at right angles.
Vectors with a dot product of zero. Perpendicular vectors meet at right angles.
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Find the projection of $\begin{bmatrix} 3 \\ 4 \matrix}$ onto $\begin{bmatrix} 6 \\ 8 \matrix}$.
Find the projection of $\begin{bmatrix} 3 \\ 4 \matrix}$ onto $\begin{bmatrix} 6 \\ 8 \matrix}$.
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$\begin{bmatrix} 3 \\ 4 \matrix}$. Both vectors are parallel, so projection equals the first vector.
$\begin{bmatrix} 3 \\ 4 \matrix}$. Both vectors are parallel, so projection equals the first vector.
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What does a negative scalar multiplication do to a vector?
What does a negative scalar multiplication do to a vector?
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Reverses the direction of the vector. Changes direction but preserves magnitude.
Reverses the direction of the vector. Changes direction but preserves magnitude.
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Find the angle between $\begin{bmatrix} 1 \\ 0 \matrix}$ and $\begin{bmatrix} 0 \\ 1 \matrix}$.
Find the angle between $\begin{bmatrix} 1 \\ 0 \matrix}$ and $\begin{bmatrix} 0 \\ 1 \matrix}$.
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$90^\text{°}$. Standard unit vectors are perpendicular to each other.
$90^\text{°}$. Standard unit vectors are perpendicular to each other.
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Find the sum of vectors $\begin{bmatrix} -2 \\ 5 \matrix}$ and $\begin{bmatrix} 7 \\ -3 \matrix}$.
Find the sum of vectors $\begin{bmatrix} -2 \\ 5 \matrix}$ and $\begin{bmatrix} 7 \\ -3 \matrix}$.
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$\begin{bmatrix} 5 \\ 2 \matrix}$. Add components: $(-2+7, 5-3) = (5,2)$.
$\begin{bmatrix} 5 \\ 2 \matrix}$. Add components: $(-2+7, 5-3) = (5,2)$.
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What is the inverse of vector $\textbf{v}$?
What is the inverse of vector $\textbf{v}$?
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The vector $-\textbf{v}$. (Negation). The additive inverse of a vector.
The vector $-\textbf{v}$. (Negation). The additive inverse of a vector.
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Find the magnitude of $\begin{bmatrix} 7 \\ 24 \matrix}$.
Find the magnitude of $\begin{bmatrix} 7 \\ 24 \matrix}$.
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$25$. Use $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = 25$.
$25$. Use $\sqrt{7^2 + 24^2} = \sqrt{49 + 576} = 25$.
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State the associative property of vector addition.
State the associative property of vector addition.
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($\textbf{u} + \textbf{v}) + \textbf{w} = \textbf{u} + (\textbf{v} + \textbf{w})$. Grouping doesn't affect vector addition results.
($\textbf{u} + \textbf{v}) + \textbf{w} = \textbf{u} + (\textbf{v} + \textbf{w})$. Grouping doesn't affect vector addition results.
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What is a linear combination of vectors?
What is a linear combination of vectors?
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A sum of scalar multiples of vectors. Combines vectors with scalar coefficients.
A sum of scalar multiples of vectors. Combines vectors with scalar coefficients.
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What does it mean for vectors to be linearly dependent?
What does it mean for vectors to be linearly dependent?
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One vector is a linear combination of the others. One vector can be expressed using the others.
One vector is a linear combination of the others. One vector can be expressed using the others.
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What operation is used to find a vector's component along another vector?
What operation is used to find a vector's component along another vector?
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Projection. Projects one vector onto the direction of another.
Projection. Projects one vector onto the direction of another.
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What is the result of adding a vector to its negative?
What is the result of adding a vector to its negative?
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The zero vector. A vector plus its additive inverse equals zero vector.
The zero vector. A vector plus its additive inverse equals zero vector.
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How do you find the angle $\theta$ between two vectors $\textbf{u}$ and $\textbf{v}$?
How do you find the angle $\theta$ between two vectors $\textbf{u}$ and $\textbf{v}$?
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$\theta = \text{acos}\bigg(\frac{\textbf{u} \bullet \textbf{v}}{|\textbf{u}||\textbf{v}|}\bigg)$. Uses the dot product formula and inverse cosine function.
$\theta = \text{acos}\bigg(\frac{\textbf{u} \bullet \textbf{v}}{|\textbf{u}||\textbf{v}|}\bigg)$. Uses the dot product formula and inverse cosine function.
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What does it mean if the dot product of two vectors is zero?
What does it mean if the dot product of two vectors is zero?
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The vectors are orthogonal (perpendicular). Zero dot product indicates 90° angle between vectors.
The vectors are orthogonal (perpendicular). Zero dot product indicates 90° angle between vectors.
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State the formula for the dot product of $\textbf{u} = \begin{bmatrix} a \\ b \matrix}$ and $\textbf{v} = \begin{bmatrix} c \\ d \matrix}$.
State the formula for the dot product of $\textbf{u} = \begin{bmatrix} a \\ b \matrix}$ and $\textbf{v} = \begin{bmatrix} c \\ d \matrix}$.
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$a \times c + b \times d$. Multiply corresponding components and sum the results.
$a \times c + b \times d$. Multiply corresponding components and sum the results.
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