Component Form:

\[ \vec{v} = \langle a, b, c \rangle \]

Unit Vector Form:

\[ \vec{v} = a\vec{i} + b\vec{j} + c\vec{k} \]

Given initial point \( P_1(x_1, y_1, z_1) \) and terminal point \( P_2(x_2, y_2, z_2) \):

\[ \vec{v} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle \]

The components represent displacement in x, y, and z directions

Find component form: Initial point (1, 2, 3), Terminal point (4, 6, 7)

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

\[ |\vec{v}| = \|\vec{v}\| = \sqrt{a^2 + b^2 + c^2} \]

This is an extension of the Pythagorean theorem to three dimensions

Find magnitude: \( \vec{v} = \langle 2, 3, 6 \rangle \)

Find magnitude: \( \vec{v} = \langle 1, 2, 2 \rangle \)

Find magnitude: \( \vec{v} = 2\vec{i} + 3\vec{j} - \vec{k} \)

Add corresponding components:

\[ \vec{u} + \vec{v} = \langle a_1, b_1, c_1 \rangle + \langle a_2, b_2, c_2 \rangle \] \[ = \langle a_1 + a_2, b_1 + b_2, c_1 + c_2 \rangle \]

\[ \vec{u} - \vec{v} = \langle a_1, b_1, c_1 \rangle - \langle a_2, b_2, c_2 \rangle \] \[ = \langle a_1 - a_2, b_1 - b_2, c_1 - c_2 \rangle \]

Add: \( \vec{u} = \langle 1, 2, 3 \rangle \) and \( \vec{v} = \langle 4, -1, 2 \rangle \)

Subtract: \( \vec{u} = \langle 6, 3, 8 \rangle \) and \( \vec{v} = \langle 2, 5, 3 \rangle \)

Multiply each component by the scalar:

\[ k\vec{v} = k\langle a, b, c \rangle = \langle ka, kb, kc \rangle \]

• Magnitude: \( |k\vec{v}| = |k| \cdot |\vec{v}| \)

• If \( k > 0 \): direction stays the same

• If \( k < 0 \): direction reverses

• If \( k = 0 \): result is zero vector \( \langle 0, 0, 0 \rangle \)

Find: \( 3\vec{v} \) where \( \vec{v} = \langle 2, -1, 4 \rangle \)

Find: \( -2\vec{v} \) where \( \vec{v} = \langle 1, 3, -2 \rangle \)

A unit vector has magnitude 1. To find the unit vector in the direction of \( \vec{v} \):

\[ \hat{v} = \frac{\vec{v}}{|\vec{v}|} = \frac{1}{|\vec{v}|}\vec{v} = \frac{\langle a, b, c \rangle}{\sqrt{a^2 + b^2 + c^2}} \]

\[ \vec{i} = \langle 1, 0, 0 \rangle \quad \vec{j} = \langle 0, 1, 0 \rangle \quad \vec{k} = \langle 0, 0, 1 \rangle \]

These are orthogonal (perpendicular) to each other

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

Find unit vector: \( \vec{v} = \langle -3, 0, 4 \rangle \)

A linear combination uses scalar multiplication and addition:

\[ \vec{w} = c_1\vec{v}_1 + c_2\vec{v}_2 + c_3\vec{v}_3 \]

where \( c_1, c_2, c_3 \) are scalars

Any 3D vector can be written as a linear combination of \( \vec{i}, \vec{j}, \vec{k} \):

\[ \vec{v} = a\vec{i} + b\vec{j} + c\vec{k} \]

Find: \( 2\vec{u} + 3\vec{v} \) where \( \vec{u} = \langle 1, 2, 3 \rangle \) and \( \vec{v} = \langle 2, -1, 1 \rangle \)

Find: \( 3\vec{i} - 2\vec{j} + 4\vec{k} \)

Find: \( 2\vec{u} - 3\vec{v} + \vec{w} \) where \( \vec{u} = \langle 1, 0, 2 \rangle \), \( \vec{v} = \langle 2, 1, -1 \rangle \), \( \vec{w} = \langle 0, 3, 1 \rangle \)