4.8 Vectors

What are vectors in AP Precalculus?

A vector is a directed line segment with both magnitude (length) and direction. In AP Precalculus you write vectors in component form like $\langle a, b \rangle$, find their magnitude and direction, and combine them using scalar multiplication, addition, dot products, and unit vectors.

Why This Matters for the AP Precalculus Exam

Vectors are part of Unit 4, which is not assessed on the AP Precalculus exam. The AP Exam covers material from Units 1, 2, and 3. Even so, this topic is worth learning because it builds skills you will use in calculus, physics, and other college courses.

Working with vectors strengthens habits that show up across the course: moving between graphical, numerical, and analytical representations, choosing the right procedure, and communicating your reasoning clearly. Vectors also lead directly into vector-valued functions and matrices later in Unit 4, so getting comfortable here makes the rest of the unit easier.

Key Takeaways

• A vector has both magnitude and direction. The starting point is the tail and the ending point is the head.
• For a vector from $P_1 = (x_1, y_1)$ to $P_2 = (x_2, y_2)$, the component form is $\langle a, b \rangle$ where $a = x_2 - x_1$ and $b = y_2 - y_1$.
• Magnitude is $\sqrt{a^2 + b^2}$, found from the components.
• Scalar multiplication scales each component and keeps the vector parallel to the original; vector addition adds matching components.
• The dot product is $\langle a_1, b_1 \rangle \cdot \langle a_2, b_2 \rangle = a_1 a_2 + b_1 b_2$, and it equals zero exactly when two nonzero vectors are perpendicular.
• A unit vector has magnitude 1 and is found by multiplying a nonzero vector by the reciprocal of its magnitude.

What a Vector Is

A vector is a directed line segment. It carries two pieces of information: how long it is (magnitude) and which way it points (direction). Vectors model quantities like velocity, force, and displacement, where direction matters as much as size.

When you draw a vector as an arrow:

• The tail is where the vector starts.
• The head is where the arrow points and ends.
• The length of the segment is the magnitude.

Representing Vectors in Components

A vector in the xy-plane is described by two components. If a vector goes from $P_1 = (x_1, y_1)$ to $P_2 = (x_2, y_2)$, then:

$a = x_2 - x_1 \qquad b = y_2 - y_1$

You write the vector as $\langle a, b \rangle$. The component $a$ is the horizontal change and $b$ is the vertical change.

The zero vector $\langle 0, 0 \rangle$ is the special case when $P_1$ and $P_2$ are the same point, so the magnitude is 0.

Direction and Magnitude

The direction of $\langle a, b \rangle$ is parallel to the line from the origin to the point $(a, b)$. The magnitude is:

$\|\langle a, b \rangle\| = \sqrt{a^2 + b^2}$

Finding Components with Trigonometry

If a vector is given geometrically by its length and the angle it makes with the x-axis, you can find its components with trigonometry. For a vector of magnitude $r$ at angle $\theta$:

$a = r\cos\theta \qquad b = r\sin\theta$

The horizontal component uses cosine and the vertical component uses sine.

Vector Operations

Scalar Multiplication

Multiplying a vector by a constant $k$ scales each component:

$k\langle a, b \rangle = \langle ka, kb \rangle$

The result is parallel to the original vector. Its magnitude becomes $|k|$ times the original magnitude. A negative $k$ reverses the direction.

Vector Addition

To add two vectors, add their matching components:

$\langle a_1, b_1 \rangle + \langle a_2, b_2 \rangle = \langle a_1 + a_2, b_1 + b_2 \rangle$

Graphically, place the tail of the second vector at the head of the first. The sum runs from the tail of the first vector to the head of the second. This is the tip-to-tail method, and the same sum is the diagonal of the parallelogram formed by the two vectors.

Dot Product

The dot product of two vectors is a single number (a scalar):

$\langle a_1, b_1 \rangle \cdot \langle a_2, b_2 \rangle = a_1 a_2 + b_1 b_2$

For example, with $A = \langle 3, 4 \rangle$ and $B = \langle 2, -1 \rangle$:

$A \cdot B = (3)(2) + (4)(-1) = 6 - 4 = 2$

The dot product also equals the product of the two magnitudes and the cosine of the angle between them:

$\vec{u} \cdot \vec{v} = \|\vec{u}\|\,\|\vec{v}\|\cos\theta$

This gives a quick perpendicularity test: if the dot product of two nonzero vectors is 0, the angle between them is 90 degrees, so the vectors are perpendicular. You can also rearrange the formula to find the angle between two vectors:

$\cos\theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\|\,\|\vec{v}\|}$

Unit Vectors

A unit vector has magnitude 1. To build a unit vector pointing the same way as a nonzero vector, multiply the vector by the reciprocal of its magnitude:

$\hat{u} = \frac{1}{\|\vec{v}\|}\,\vec{v}$

For example, if a vector has magnitude 5, multiply it by $\frac{1}{5}$ to shrink it to length 1 while keeping its direction.

Any vector $\langle a, b \rangle$ can also be written using the standard unit vectors $\vec{i}$ and $\vec{j}$:

$\langle a, b \rangle = a\vec{i} + b\vec{j}$

where $\vec{i} = \langle 1, 0 \rangle$ points along the x-axis and $\vec{j} = \langle 0, 1 \rangle$ points along the y-axis.

Triangles Formed by Vector Addition

When you add vectors, the vectors and their sum can form a triangle. The Law of Sines and the Law of Cosines help you find unknown side lengths and angle measures in those triangles.

Law of Sines

For a triangle with sides $a$, $b$, $c$ and opposite angles $A$, $B$, $C$:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Law of Cosines

$a^2 = b^2 + c^2 - 2bc\cos A$

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

Use the Law of Cosines when you know two sides and the included angle, or all three sides. Use the Law of Sines when you have an angle paired with its opposite side. Both are useful for analyzing the triangle made by two vectors and their resultant.

How to Use This on the AP Precalculus Exam

Unit 4 is not tested on the AP Precalculus exam, so you will not see vector questions on the official test. The practice below helps you build fluency for class assessments and later courses.

Problem Solving

• Read carefully whether a vector is given by two endpoints or by a magnitude and angle. Use $a = x_2 - x_1$, $b = y_2 - y_1$ for endpoints, and $a = r\cos\theta$, $b = r\sin\theta$ for magnitude and angle.
• Keep components in order. Mixing up $x$ and $y$ changes both the magnitude and the direction.
• For perpendicularity, compute the dot product first. If it is 0, the vectors are perpendicular, and you do not need to find the actual angle.
• When finding a unit vector, compute the magnitude first, then multiply each component by its reciprocal. Check your answer by confirming the new magnitude is 1.

Common Trap

• A scalar times a vector is still a vector, but a dot product of two vectors is a scalar. Watch what type of object your answer should be.
• Magnitude is always nonnegative because of the square root, even when components are negative.

Common Misconceptions

• A vector is not just its length. Two vectors with the same magnitude can point in different directions, which makes them different vectors.
• Component order matters. $\langle a, b \rangle$ means horizontal change $a$ and vertical change $b$. Swapping them gives a different vector.
• The dot product is a number, not a vector. Its sign tells you whether vectors point in generally similar or opposite directions, and a value of 0 means perpendicular.
• A zero dot product does not mean a zero vector. It means the two vectors are perpendicular, as long as neither one is the zero vector.
• Scaling by a negative number flips direction. The result stays parallel to the original, but it points the opposite way.
• Unit vector does not mean rounding to length 1. You divide by the exact magnitude, so the components are usually not whole numbers.

Related AP Precalculus Guides

• 4.10 Matrices
• 4.12 Linear Transformations and Matrices
• 4.5 Implicitly Defined Functions
• 4.9 Vector-Valued Functions
• 4.13 Matrices as Functions
• 4.4 Parametrically Defined Circles and Lines