AP Precalculus Vectors — AP Precalculus
1. Core Vector Definitions ★★☆☆☆ ⏱ 3 min
A vector is a mathematical quantity with both magnitude (size) and direction, unlike a scalar, which only has magnitude. Common examples tested on AP Precalculus include displacement, velocity, and force. Vectors account for ~3-4% of your total AP Precalculus exam score, appearing in both multiple-choice and free-response sections.
2. Components, Magnitude, and Direction ★★★☆☆ ⏱ 4 min
Any planar vector can be split into horizontal ($x$) and vertical ($y$) components, which give displacement along each axis. For a vector from initial point $P(x_1,y_1)$ to terminal point $Q(x_2,y_2)$, component form is calculated as terminal minus initial coordinates:
\vec{v} = \langle x_2 - x_1, y_2 - y_1 \rangle
Magnitude (length) of $\vec{v} = \langle a, b \rangle$ comes from the Pythagorean theorem:
|\vec{v}| = \sqrt{a^2 + b^2}
Direction is given as the standard position angle $\theta$, measured counterclockwise from the positive $x$-axis. To find $\theta$ from components: $\tan\theta = \frac{b}{a}$, so $\theta = \arctan\left(\frac{b}{a}\right)$ plus quadrant correction: add $180^\circ$ (or $\pi$ radians) for Quadrants II/III, add $360^\circ$ for negative angles in Quadrant IV. If given magnitude $r=|\vec{v}|$ and direction $\theta$, components are $a = r\cos\theta, b = r\sin\theta$.
Exam tip: Always sketch a quick rough plot of your vector on a coordinate grid before reporting the direction angle to catch quadrant correction mistakes.
3. Vector Addition and Scalar Multiplication ★★★☆☆ ⏱ 4 min
Vector addition produces a resultant vector from two input vectors. Algebraically, addition is always component-wise for vectors in component form:
\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2 \rangle
Scalar multiplication multiplies a vector by a real scalar $k$, scaling its magnitude and reversing direction if $k$ is negative. This is also component-wise:
k\vec{u} = \langle k u_1, k u_2 \rangle
A unit vector is a vector with magnitude 1, used to represent direction only. To get a unit vector in the direction of non-zero $\vec{v}$, divide by its magnitude: $\hat{v} = \frac{1}{|\vec{v}|}\vec{v}$. The standard unit vectors are $\hat{i} = \langle 1, 0 \rangle$ and $\hat{j} = \langle 0, 1 \rangle$, so any vector $\langle a, b \rangle$ can be written as $a\hat{i} + b\hat{j}$.
Exam tip: Rewrite subtraction of scalar multiples as addition of the negative scalar multiple, like $2\vec{u} - 3\vec{v} = 2\vec{u} + (-3)\vec{v}$, to avoid sign errors when distributing.
4. Dot Product and Vector Projections ★★★★☆ ⏱ 4 min
The dot product (or scalar product) is an operation that takes two vectors and returns a scalar (not a vector). For $\vec{u} = \langle u_1, u_2 \rangle$ and $\vec{v} = \langle v_1, v_2 \rangle$:
\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2
The dot product relates to the angle $\theta$ (between $0^\circ$ and $180^\circ$) between two vectors via the identity:
\vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|\cos\theta
Two key uses on the AP exam are: 1) testing for orthogonality (perpendicularity): vectors are orthogonal if and only if their dot product is zero; 2) finding the projection of one vector onto another, which is the shadow of $\vec{u}$ along the line containing $\vec{v}$. The formula for the vector projection of $\vec{u}$ onto $\vec{v}$ is:
\text{proj}_{\vec{v}} \vec{u} = \left(\frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2}\right)\vec{v}
Exam tip: Remember that vector projection uses $|\vec{v}|^2$ in the denominator, while scalar projection uses $|\vec{v}|$. Write both formulas down before solving to avoid mixing them up.
5. AP-Style Concept Check ★★★☆☆ ⏱ 3 min
Common Pitfalls
Why: Arctangent only outputs values between $-90^\circ$ and $90^\circ$, so it does not automatically account for vectors in Quadrants II and III.
Why: Students often forget to distribute the negative sign to both components of $3\vec{v}$ when subtracting.
Why: Students confuse the dot product with component-wise multiplication.
Why: Students mix up the order of subtraction for initial vs terminal points.
Why: Students mix up the formulas for scalar projection and vector projection.
Why: Students forget the full definition of equal vectors.