AP®︎ Precalculus: Topic 4.8 Practice Test

Essential Knowledge: 4.8.A.1

A vector is a directed line segment. When a vector is placed in the plane, the point at the beginning of the line segment is called the tail, and the point at the end of the line segment is called the head. The length of the line segment is the magnitude of the vector.

Essential Knowledge: 4.8.A.2

A vector $\overrightarrow{P_1P_2}$ with two components can be plotted in the $xy$-plane from $P_1=(x_1,y_1)$ to $P_2=(x_2,y_2)$. The vector is identified by $a$ and $b$, where $a=x_2-x_1$ and $b=y_2-y_1$. The vector can be expressed as $\langle a,b\rangle$. A zero vector $\langle 0,0\rangle$ is the trivial case when $P_1=P_2$.

Essential Knowledge: 4.8.A.3

The direction of the vector is parallel to the line segment from the origin to the point with coordinates $(a,b)$. The magnitude of the vector is the square root of the sum of the squares of the components.

Essential Knowledge: 4.8.A.4

For a vector represented geometrically in the plane, the components of the vector can be found using trigonometry.

Essential Knowledge: 4.8.B.1

The multiplication of a constant and a vector results in a new vector whose components are found by multiplying the constant by each of the components of the original vector. The new vector is parallel to the original vector.

Essential Knowledge: 4.8.B.2

The sum of two vectors in $\mathbb{R}^2$ is a new vector whose components are found by adding the corresponding components of the original vectors. The new vector can be represented graphically as a vector whose tail corresponds to the tail of the first vector and whose head corresponds to the head of the second vector when the second vector’s tail is located at the first vector’s head.

Essential Knowledge: 4.8.B.3

The dot product of two vectors is the sum of the products of their corresponding components. That is, $\langle a_1,b_1\rangle\cdot\langle a_2,b_2\rangle=a_1a_2+b_1b_2$.

Essential Knowledge: 4.8.C.1

A unit vector is a vector of magnitude 1. A unit vector in the same direction as a given nonzero vector can be found by scalar multiplying the vector by the reciprocal of its magnitude.

Essential Knowledge: 4.8.C.2

The vector $\langle a,b\rangle$ can be expressed as $a\vec{i}+b\vec{j}$ in $\mathbb{R}^2$, where $\vec{i}$ and $\vec{j}$ are unit vectors in the $x$ and $y$ directions, respectively. That is, $\vec{i}=\langle 1,0\rangle$ and $\vec{j}=\langle 0,1\rangle$.

Essential Knowledge: 4.8.D.1

The dot product is geometrically equivalent to the product of the magnitudes of the two vectors and the cosine of the angle between them. Therefore, if the dot product of two nonzero vectors is zero, then the vectors are perpendicular.

Essential Knowledge: 4.8.D.2

The Law of Sines and Law of Cosines can be used to determine side lengths and angle measures of triangles formed by vector addition.