Essential Vector Operations

Why This Matters

Vectors are the language of motion, force, and space—and you'll encounter them constantly throughout calculus and beyond. When you're working with position, velocity, and acceleration in multivariable calculus, or analyzing work, torque, and flux in physics applications, you're using vector operations. The AP exam tests whether you understand how vectors combine, how they measure relationships between directions, and how they describe geometric objects in space.

Don't just memorize formulas—know what each operation does conceptually. Can you explain why the dot product tells you about angles? Why the cross product gives you area? These conceptual connections are exactly what FRQ prompts target. Master the underlying geometry, and the calculations become intuitive.

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Combining and Scaling Vectors

These foundational operations let you build new vectors from existing ones. Addition combines effects; scalar multiplication stretches or shrinks without changing direction (unless you flip it).

Vector Addition and Subtraction

• Component-wise combination—add corresponding components: $\vec{u} + \vec{v} = \langle u_1 + v_1, u_2 + v_2, u_3 + v_3 \rangle$
• Geometric interpretation uses the parallelogram law or tip-to-tail method to visualize the resultant
• Subtraction means adding the negative: $\vec{u} - \vec{v} = \vec{u} + (-\vec{v})$, pointing from $\vec{v}$'s tip to $\vec{u}$'s tip

Scalar Multiplication

• Scaling magnitude—multiplying by scalar $c$ gives $c\vec{v}$ with magnitude $|c||\vec{v}|$
• Direction preserved when $c > 0$; direction reversed when $c < 0$
• Zero scalar produces the zero vector regardless of the original vector's magnitude

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Measuring Vectors

Before you can work with vectors effectively, you need to quantify their size and standardize their direction. Magnitude gives length; unit vectors give pure direction.

Magnitude (Length) of a Vector

• Pythagorean extension—for $\vec{v} = \langle x, y, z \rangle$, magnitude is $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$
• Always non-negative—magnitude equals zero only for the zero vector
• Distance formula connection—the magnitude of $\vec{PQ}$ gives the distance between points $P$ and $Q$

Unit Vectors

• Magnitude of exactly 1—obtained by dividing any nonzero vector by its magnitude: $\hat{v} = \frac{\vec{v}}{|\vec{v}|}$
• Standard basis vectors $\hat{i}, \hat{j}, \hat{k}$ point along the positive $x$-, $y$-, and $z$-axes
• Direction isolation—unit vectors let you separate how far from which way

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Products That Reveal Geometric Relationships

The dot and cross products are your primary tools for measuring angles, checking perpendicularity, and finding areas. The dot product outputs a scalar; the cross product outputs a vector.

Dot Product (Scalar Product)

• Two equivalent formulas: $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3$ or $\vec{u} \cdot \vec{v} = |\vec{u}||\vec{v}|\cos\theta$
• Angle finder—rearrange to get $\cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}||\vec{v}|}$
• Orthogonality test—vectors are perpendicular if and only if $\vec{u} \cdot \vec{v} = 0$

Cross Product (Vector Product)

• Produces a perpendicular vector—$\vec{u} \times \vec{v}$ is orthogonal to both $\vec{u}$ and $\vec{v}$
• Determinant calculation: $\vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$
• Magnitude equals parallelogram area: $|\vec{u} \times \vec{v}| = |\vec{u}||\vec{v}|\sin\theta$

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Breaking Vectors Apart

Sometimes you need to analyze how much of a vector points in a particular direction. Projection extracts the component along another vector; decomposition separates into coordinate directions.

Vector Projection

• Formula: $\text{proj}_{\vec{v}}\vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2}\vec{v}$ gives the component of $\vec{u}$ in the direction of $\vec{v}$
• Scalar projection (signed length) is $\text{comp}_{\vec{v}}\vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|}$
• Physics application—resolving force into components parallel and perpendicular to a surface

Vector Decomposition

• Component form—any vector $\vec{v} = \langle a, b, c \rangle$ decomposes as $a\hat{i} + b\hat{j} + c\hat{k}$
• Arbitrary directions—you can decompose along any set of basis vectors, not just coordinate axes
• Problem-solving strategy—choose axes aligned with the geometry of your problem to simplify calculations

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Describing Geometric Objects

Vectors provide elegant equations for lines and planes in 3D space. A line needs a point and a direction; a plane needs a point and a normal.

Vector Equations of Lines and Planes

• Line equation: $\vec{r}(t) = \vec{r}_0 + t\vec{v}$ where $\vec{r}_0$ is a point and $\vec{v}$ is the direction vector
• Plane equation: $\vec{n} \cdot (\vec{r} - \vec{r}_0) = 0$ where $\vec{n}$ is the normal vector
• Parametric conversion—vector equations easily convert to parametric form for calculations

Vector-Valued Functions

• Maps scalars to vectors—$\vec{r}(t) = \langle f(t), g(t), h(t) \rangle$ traces a curve as $t$ varies
• Derivatives give velocity—$\vec{r}'(t)$ is tangent to the curve and represents instantaneous velocity
• Integrals recover position—integrating acceleration gives velocity; integrating velocity gives position

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Quick Reference Table

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Self-Check Questions

1. Which two operations both produce scalar outputs, and how do their geometric interpretations differ?

2. If $\vec{u} \cdot \vec{v} = 0$ and $\vec{u} \times \vec{v} = \vec{0}$, what can you conclude about vectors $\vec{u}$ and $\vec{v}$?

3. Compare and contrast: How would you use the dot product versus the cross product to solve a problem involving the angle between two vectors?

4. A vector-valued function $\vec{r}(t)$ describes a particle's position. What operation gives the particle's speed at time $t$, and what combination of concepts does this require?

5. An FRQ gives you vectors $\vec{u}$ and $\vec{v}$ and asks for the area of the triangle they form. Which operation do you use, and what adjustment must you make to get triangle area instead of parallelogram area?