AP Precalculus: Three-Dimensional Vectors Formulas & Properties
1. Magnitude of a 3D Vector
- For \( \vec{v} = \langle a, b, c \rangle \):
- \( |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \)
2. Component Form from Points
- Given points \( P(x_1, y_1, z_1) \), \( Q(x_2, y_2, z_2) \):
- \( \vec{PQ} = \langle x_2-x_1,\, y_2-y_1,\, z_2-z_1 \rangle \)
3. Vector Addition & Subtraction
- \( \langle a, b, c \rangle + \langle d, e, f \rangle = \langle a+d,\, b+e,\, c+f \rangle \)
- \( \langle a, b, c \rangle - \langle d, e, f \rangle = \langle a-d,\, b-e,\, c-f \rangle \)
4. Scalar Multiplication
- \( k \langle a, b, c \rangle = \langle ka,\, kb,\, kc \rangle \)
- \( |k \vec{v}| = |k| |\vec{v}| \)
5. Unit Vector in Same Direction
- For \( \vec{v} \neq 0 \):
\( \mathbf{u} = \frac{\vec{v}}{|\vec{v}|} \) - \( \mathbf{u} \) has magnitude 1 and same direction as \( \vec{v} \)
6. Linear Combinations
- \( a \vec{u} + b \vec{w} = a\langle u_1, u_2, u_3 \rangle + b\langle w_1, w_2, w_3 \rangle = \langle a u_1 + b w_1,\, a u_2 + b w_2,\, a u_3 + b w_3 \rangle \)