Vectors - ACT Math
Card 1 of 30
What operation finds the angle between $\mathbf{a}$ and $\mathbf{b}$ if they are not zero vectors?
What operation finds the angle between $\mathbf{a}$ and $\mathbf{b}$ if they are not zero vectors?
Tap to reveal answer
Dot product. Cosine formula requires dot product calculation.
Dot product. Cosine formula requires dot product calculation.
← Didn't Know|Knew It →
Calculate the cross product of $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$.
Calculate the cross product of $\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$.
Tap to reveal answer
$\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$. Use the right-hand rule: $\textbf{i} \times \textbf{j} = \textbf{k}$.
$\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$. Use the right-hand rule: $\textbf{i} \times \textbf{j} = \textbf{k}$.
← Didn't Know|Knew It →
What is the angle between vectors $\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}$?
What is the angle between vectors $\begin{bmatrix} 1 \\ 0 \\ \end{bmatrix}$ and $\begin{bmatrix} 0 \\ 1 \\ \end{bmatrix}$?
Tap to reveal answer
$90^\circ$. These are perpendicular unit vectors along coordinate axes.
$90^\circ$. These are perpendicular unit vectors along coordinate axes.
← Didn't Know|Knew It →
Find the unit vector for $\textbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.
Find the unit vector for $\textbf{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$.
Tap to reveal answer
$\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Divide by magnitude: $|\textbf{v}| = \sqrt{3^2 + 4^2} = 5$.
$\begin{bmatrix} \frac{3}{5} \\ \frac{4}{5} \end{bmatrix}$. Divide by magnitude: $|\textbf{v}| = \sqrt{3^2 + 4^2} = 5$.
← Didn't Know|Knew It →
Find the vector from point $A(1,2)$ to point $B(4,6)$.
Find the vector from point $A(1,2)$ to point $B(4,6)$.
Tap to reveal answer
$$\begin{bmatrix} 3 \\ 4 \end{bmatrix}$$ Subtract starting point coordinates from ending point coordinates.
$$\begin{bmatrix} 3 \\ 4 \end{bmatrix}$$ Subtract starting point coordinates from ending point coordinates.
← Didn't Know|Knew It →
What is the vector $\mathbf{v}$ if $\mathbf{v} = \langle 0, 0 \rangle$?
What is the vector $\mathbf{v}$ if $\mathbf{v} = \langle 0, 0 \rangle$?
Tap to reveal answer
Zero vector. Vector with no magnitude or direction.
Zero vector. Vector with no magnitude or direction.
← Didn't Know|Knew It →
What is the midpoint of $A(-2,5)$ and $B(4,1)$?
What is the midpoint of $A(-2,5)$ and $B(4,1)$?
Tap to reveal answer
$\left(1,;3\right)$. $\left(\frac{-2+4}{2},\frac{5+1}{2}\right)=(1,3)$.
$\left(1,;3\right)$. $\left(\frac{-2+4}{2},\frac{5+1}{2}\right)=(1,3)$.
← Didn't Know|Knew It →
What is the formula for the magnitude of a 3D vector $\mathbf{v} = \langle a, b, c \rangle$?
What is the formula for the magnitude of a 3D vector $\mathbf{v} = \langle a, b, c \rangle$?
Tap to reveal answer
$\sqrt{a^2 + b^2 + c^2}$. 3D extension of Pythagorean theorem.
$\sqrt{a^2 + b^2 + c^2}$. 3D extension of Pythagorean theorem.
← Didn't Know|Knew It →
What is the associative property of scalar multiplication with vectors?
What is the associative property of scalar multiplication with vectors?
Tap to reveal answer
$(ab)\mathbf{v} = a(b\mathbf{v})$. Scalar multiplication order doesn't matter.
$(ab)\mathbf{v} = a(b\mathbf{v})$. Scalar multiplication order doesn't matter.
← Didn't Know|Knew It →
Find the unit vector along $\mathbf{v} = \langle 1, 1 \rangle$.
Find the unit vector along $\mathbf{v} = \langle 1, 1 \rangle$.
Tap to reveal answer
$\frac{1}{\sqrt{2}} \langle 1, 1 \rangle$. Magnitude is $\sqrt{2}$, so divide by it.
$\frac{1}{\sqrt{2}} \langle 1, 1 \rangle$. Magnitude is $\sqrt{2}$, so divide by it.
← Didn't Know|Knew It →
How do you calculate the direction angle of vector $\textbf{v} = \begin{bmatrix} a \\ b \end{bmatrix}$?
How do you calculate the direction angle of vector $\textbf{v} = \begin{bmatrix} a \\ b \end{bmatrix}$?
Tap to reveal answer
$\theta = \tan^{-1}(\frac{b}{a})$. Use arctangent of the ratio of vertical to horizontal components.
$\theta = \tan^{-1}(\frac{b}{a})$. Use arctangent of the ratio of vertical to horizontal components.
← Didn't Know|Knew It →
Identify the scalar multiplication property for vector $\textbf{v}$ and scalar $0$.
Identify the scalar multiplication property for vector $\textbf{v}$ and scalar $0$.
Tap to reveal answer
Result is the zero vector. Multiplying any vector by zero gives the zero vector.
Result is the zero vector. Multiplying any vector by zero gives the zero vector.
← Didn't Know|Knew It →
Determine if vectors $\begin{bmatrix} 3 \\ 6 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 4 \end{bmatrix}$ are parallel.
Determine if vectors $\begin{bmatrix} 3 \\ 6 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ 4 \end{bmatrix}$ are parallel.
Tap to reveal answer
Yes, they are parallel. Check if $\frac{3}{2} = \frac{6}{4}$, which simplifies to $\frac{3}{2} = \frac{3}{2}$.
Yes, they are parallel. Check if $\frac{3}{2} = \frac{6}{4}$, which simplifies to $\frac{3}{2} = \frac{3}{2}$.
← Didn't Know|Knew It →
Determine if vectors $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$ are orthogonal.
Determine if vectors $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$ are orthogonal.
Tap to reveal answer
No, they are not orthogonal. Dot product is $1(4) + 2(5) + 3(6) = 32 \neq 0$.
No, they are not orthogonal. Dot product is $1(4) + 2(5) + 3(6) = 32 \neq 0$.
← Didn't Know|Knew It →
Given $\textbf{a} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, find $-2\textbf{a}$.
Given $\textbf{a} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$, find $-2\textbf{a}$.
Tap to reveal answer
$\begin{bmatrix} -6 \\ -8 \end{bmatrix}$. Multiply each component by the scalar $-2$.
$\begin{bmatrix} -6 \\ -8 \end{bmatrix}$. Multiply each component by the scalar $-2$.
← Didn't Know|Knew It →
What is the vector projection of $\textbf{a}$ onto $\textbf{b}$?
What is the vector projection of $\textbf{a}$ onto $\textbf{b}$?
Tap to reveal answer
$\frac{\textbf{a} \bullet \textbf{b}}{\text{||b||}^2} \textbf{b}$. Projects vector $\textbf{a}$ onto the direction of vector $\textbf{b}$.
$\frac{\textbf{a} \bullet \textbf{b}}{\text{||b||}^2} \textbf{b}$. Projects vector $\textbf{a}$ onto the direction of vector $\textbf{b}$.
← Didn't Know|Knew It →
State the result of the cross product of parallel vectors.
State the result of the cross product of parallel vectors.
Tap to reveal answer
The zero vector. Cross product of parallel vectors always equals zero.
The zero vector. Cross product of parallel vectors always equals zero.
← Didn't Know|Knew It →
Calculate the magnitude of vector $\begin{bmatrix} 5 \\ 12 \end{bmatrix}$.
Calculate the magnitude of vector $\begin{bmatrix} 5 \\ 12 \end{bmatrix}$.
Tap to reveal answer
$13$. Calculate $\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
$13$. Calculate $\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
← Didn't Know|Knew It →
How do you find the unit vector of $\textbf{v} = \begin{bmatrix} a \\ b \\ \end{bmatrix}$?
How do you find the unit vector of $\textbf{v} = \begin{bmatrix} a \\ b \\ \end{bmatrix}$?
Tap to reveal answer
$\frac{\textbf{v}}{||v||} = \begin{bmatrix} \frac{a}{||v||} \\ \frac{b}{||v||} \\ \end{bmatrix}$. Divide the vector by its magnitude to get length 1.
$\frac{\textbf{v}}{||v||} = \begin{bmatrix} \frac{a}{||v||} \\ \frac{b}{||v||} \\ \end{bmatrix}$. Divide the vector by its magnitude to get length 1.
← Didn't Know|Knew It →
Find the magnitude of $\textbf{v} = \begin{bmatrix} -7 \\ 24 \end{bmatrix}$
Find the magnitude of $\textbf{v} = \begin{bmatrix} -7 \\ 24 \end{bmatrix}$
Tap to reveal answer
$25$. Calculate $\sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$
$25$. Calculate $\sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$
← Didn't Know|Knew It →
Express vector $\textbf{v} = \begin{bmatrix} 4 \\ -3 \end{bmatrix}$ in terms of unit vectors $\textbf{i}$ and $\textbf{j}$.
Express vector $\textbf{v} = \begin{bmatrix} 4 \\ -3 \end{bmatrix}$ in terms of unit vectors $\textbf{i}$ and $\textbf{j}$.
Tap to reveal answer
$4\textbf{i} - 3\textbf{j}$. Express using standard unit vectors $\textbf{i}$ and $\textbf{j}$.
$4\textbf{i} - 3\textbf{j}$. Express using standard unit vectors $\textbf{i}$ and $\textbf{j}$.
← Didn't Know|Knew It →
Find the dot product of $\begin{bmatrix} 2 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \\ 1 \end{bmatrix}$.
Find the dot product of $\begin{bmatrix} 2 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \\ 1 \end{bmatrix}$.
Tap to reveal answer
$11$. Calculate $(2)(4) + (3)(1) = 8 + 3 = 11$.
$11$. Calculate $(2)(4) + (3)(1) = 8 + 3 = 11$.
← Didn't Know|Knew It →
Express $\textbf{v} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ in terms of unit vectors.
Express $\textbf{v} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}$ in terms of unit vectors.
Tap to reveal answer
$\textbf{k}$. This is the standard unit vector in the z-direction.
$\textbf{k}$. This is the standard unit vector in the z-direction.
← Didn't Know|Knew It →
What is the result of scalar multiplication $k \times \begin{bmatrix} a \\ b \\ \end{bmatrix}$?
What is the result of scalar multiplication $k \times \begin{bmatrix} a \\ b \\ \end{bmatrix}$?
Tap to reveal answer
$\begin{bmatrix} ka \\ kb \\ \end{bmatrix}$. Multiply the scalar by each component.
$\begin{bmatrix} ka \\ kb \\ \end{bmatrix}$. Multiply the scalar by each component.
← Didn't Know|Knew It →
Calculate the dot product of $\begin{bmatrix} -2 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} 3 \\ -4 \end{bmatrix}$.
Calculate the dot product of $\begin{bmatrix} -2 \\ 1 \end{bmatrix}$ and $\begin{bmatrix} 3 \\ -4 \end{bmatrix}$.
Tap to reveal answer
$-10$. Calculate $(-2)(3) + (1)(-4) = -6 - 4 = -10$.
$-10$. Calculate $(-2)(3) + (1)(-4) = -6 - 4 = -10$.
← Didn't Know|Knew It →
What is the result of adding the zero vector to any vector $\textbf{v}$?
What is the result of adding the zero vector to any vector $\textbf{v}$?
Tap to reveal answer
The vector $\textbf{v}$ itself. Zero vector is the additive identity for vector addition.
The vector $\textbf{v}$ itself. Zero vector is the additive identity for vector addition.
← Didn't Know|Knew It →
What is the dot product of vectors $\textbf{u} = \begin{bmatrix} a \\ b \end{bmatrix}$ and $\textbf{v} = \begin{bmatrix} c \\ d \end{bmatrix}$?
What is the dot product of vectors $\textbf{u} = \begin{bmatrix} a \\ b \end{bmatrix}$ and $\textbf{v} = \begin{bmatrix} c \\ d \end{bmatrix}$?
Tap to reveal answer
$\textbf{u} \bullet \textbf{v} = ac + bd$. Multiply corresponding components and sum the results.
$\textbf{u} \bullet \textbf{v} = ac + bd$. Multiply corresponding components and sum the results.
← Didn't Know|Knew It →
What is the magnitude of the zero vector?
What is the magnitude of the zero vector?
Tap to reveal answer
Zero. The zero vector has no length by definition.
Zero. The zero vector has no length by definition.
← Didn't Know|Knew It →
State the condition for two vectors to be orthogonal.
State the condition for two vectors to be orthogonal.
Tap to reveal answer
Their dot product is zero. Orthogonal vectors have perpendicular direction.
Their dot product is zero. Orthogonal vectors have perpendicular direction.
← Didn't Know|Knew It →
What is the geometric interpretation of the dot product?
What is the geometric interpretation of the dot product?
Tap to reveal answer
The product of magnitudes and cosine of the angle between. Relates to both magnitude and directional alignment.
The product of magnitudes and cosine of the angle between. Relates to both magnitude and directional alignment.
← Didn't Know|Knew It →