AP Precalculus: Three-Dimensional Vectors

Extend your vector knowledge into 3D space with magnitude, components, and operations

📐 3D Magnitude 🎯 Components ➕ Operations 📏 Unit Vectors

📚 Vectors in Three Dimensions

Three-dimensional vectors extend 2D concepts by adding a z-component. While 2D vectors lie in a plane, 3D vectors exist in space and are essential for physics, engineering, and computer graphics applications like modeling motion, forces, and 3D rendering.

1 3D Vector Notation & Basics

A three-dimensional vector has three components: \(\vec{v} = \langle a, b, c \rangle\), representing movement along the x, y, and z axes.

Component Form

\(\vec{v} = \langle a, b, c \rangle\)

x-component

\(a\)

y-component

\(b\)

z-component

\(c\)

Standard Unit Vectors in 3D

\(\mathbf{i}\)

\(\langle 1, 0, 0 \rangle\)

\(\mathbf{j}\)

\(\langle 0, 1, 0 \rangle\)

\(\mathbf{k}\)

\(\langle 0, 0, 1 \rangle\)

💡 Alternative Notation

\(\vec{v} = \langle a, b, c \rangle = a\mathbf{i} + b\mathbf{j} + c\mathbf{k}\). The \(\mathbf{k}\) vector is the new unit vector along the z-axis.

2 Component Form from Two Points

Find the vector from point P to point Q by subtracting corresponding coordinates.

Vector from P to Q \(\vec{PQ} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle\)

where \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\)

📌 Example

Find: Vector from \(P(1, -2, 3)\) to \(Q(4, 5, -1)\)

Calculate: \(\vec{PQ} = \langle 4-1, 5-(-2), -1-3 \rangle = \langle 3, 7, -4 \rangle\)

3 Magnitude of a 3D Vector

The magnitude of a 3D vector is calculated using an extended version of the Pythagorean theorem.

3D Magnitude Formula \(|\vec{v}| = |\langle a, b, c \rangle| = \sqrt{a^2 + b^2 + c^2}\)

2D Magnitude

\(\sqrt{a^2 + b^2}\)

3D Magnitude

\(\sqrt{a^2 + b^2 + c^2}\)

📌 Example

Find: \(|\langle 2, -3, 6 \rangle|\)

Calculate: \(\sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7\)

4 Distance Between Two Points in 3D

The distance between two points in 3D is the magnitude of the vector connecting them.

3D Distance Formula \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\)

📌 Example

Find distance from: \(A(1, 2, 3)\) to \(B(4, 6, 3)\)

Calculate: \(d = \sqrt{(4-1)^2 + (6-2)^2 + (3-3)^2} = \sqrt{9 + 16 + 0} = \sqrt{25} = 5\)

5 Vector Addition & Subtraction

Add or subtract 3D vectors by combining their corresponding components, just like in 2D.

Addition

\(\langle a,b,c \rangle + \langle d,e,f \rangle = \langle a+d, b+e, c+f \rangle\)

Subtraction

\(\langle a,b,c \rangle - \langle d,e,f \rangle = \langle a-d, b-e, c-f \rangle\)

📌 Example

Add: \(\langle 2, -1, 4 \rangle + \langle 3, 5, -2 \rangle\)

Sum: \(\langle 2+3, -1+5, 4+(-2) \rangle = \langle 5, 4, 2 \rangle\)

6 Scalar Multiplication

Scalar multiplication multiplies each component by the scalar constant.

Scalar Multiplication \(k\langle a, b, c \rangle = \langle ka, kb, kc \rangle\)

Magnitude Effect

\(|k\vec{v}| = |k| \cdot |\vec{v}|\)

Direction Effect

Same if \(k > 0\), opposite if \(k < 0\)

📌 Example

Calculate: \(-3\langle 1, -2, 4 \rangle = \langle -3, 6, -12 \rangle\)

7 Unit Vector in 3D

A unit vector has magnitude 1 and points in the same direction as the original vector.

Unit Vector Formula \(\hat{u} = \frac{\vec{v}}{|\vec{v}|} = \frac{1}{|\vec{v}|}\langle a, b, c \rangle = \left\langle \frac{a}{|\vec{v}|}, \frac{b}{|\vec{v}|}, \frac{c}{|\vec{v}|} \right\rangle\)

📌 Example

Find unit vector in direction of: \(\vec{v} = \langle 2, -3, 6 \rangle\)

Magnitude: \(|\vec{v}| = \sqrt{4 + 9 + 36} = 7\)

Unit vector: \(\hat{u} = \frac{1}{7}\langle 2, -3, 6 \rangle = \left\langle \frac{2}{7}, -\frac{3}{7}, \frac{6}{7} \right\rangle\)

💡 Verify Your Answer

Check: \(\left(\frac{2}{7}\right)^2 + \left(-\frac{3}{7}\right)^2 + \left(\frac{6}{7}\right)^2 = \frac{4+9+36}{49} = \frac{49}{49} = 1\) ✓

8 Linear Combinations

A linear combination combines scalar multiples of multiple vectors.

Linear Combination \(a\vec{u} + b\vec{w} = \langle au_1 + bw_1, au_2 + bw_2, au_3 + bw_3 \rangle\)

📌 Example

Calculate: \(2\langle 1, 0, 3 \rangle + 3\langle -1, 2, 1 \rangle\)

= \(\langle 2, 0, 6 \rangle + \langle -3, 6, 3 \rangle\)

= \(\langle 2-3, 0+6, 6+3 \rangle = \langle -1, 6, 9 \rangle\)

9 Comparing 2D and 3D Vectors

Most 2D vector operations extend naturally to 3D by adding a third component.

Operation	2D	3D
Component Form	\(\langle a, b \rangle\)	\(\langle a, b, c \rangle\)
Magnitude	\(\sqrt{a^2 + b^2}\)	\(\sqrt{a^2 + b^2 + c^2}\)
Unit Vectors	\(\mathbf{i}, \mathbf{j}\)	\(\mathbf{i}, \mathbf{j}, \mathbf{k}\)
Addition	\(\langle a+c, b+d \rangle\)	\(\langle a+d, b+e, c+f \rangle\)
Scalar Multiply	\(\langle ka, kb \rangle\)	\(\langle ka, kb, kc \rangle\)

📋 Quick Reference

3D Magnitude

\(\sqrt{a^2 + b^2 + c^2}\)

3D Distance

\(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\)

Unit Vectors

\(\mathbf{i}, \mathbf{j}, \mathbf{k}\)

Unit Vector

\(\hat{u} = \frac{\vec{v}}{|\vec{v}|}\)

Scalar Multiply

\(k\langle a,b,c \rangle = \langle ka, kb, kc \rangle\)

Vector Addition

\(\langle a+d, b+e, c+f \rangle\)

📚 Vectors in Three Dimensions

1 3D Vector Notation & Basics

Standard Unit Vectors in 3D

2 Component Form from Two Points

3 Magnitude of a 3D Vector

2D Magnitude

3D Magnitude

4 Distance Between Two Points in 3D

5 Vector Addition & Subtraction

6 Scalar Multiplication

Magnitude Effect

7 Unit Vector in 3D

8 Linear Combinations

9 Comparing 2D and 3D Vectors

📋 Quick Reference

3D Magnitude

3D Distance

Unit Vectors

Unit Vector

Scalar Multiply

Vector Addition

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