3D Vectors

In what follows are various notes and equations dealing with 3D vectors.

Magnitude

\(\lvert a \rvert = \sqrt{a_x^2 + a_y^2 + a_z^2}\)

Distance Between Two Points

\(\lvert b - a \rvert = \sqrt{(b_x - a_x)^2 + (b_y - a_y)^2 + (b_z - a_z)^2}\)

Dot Product

\(a \cdot b = a_xb_x + a_yb_y + a_zb_z\)

Notes:

• \(a \cdot b = 0\) iff \(a\) and \(b\) are perpendicular
• \(a \cdot b = \lvert a \rvert\lvert b \rvert\) iff \(a\) and \(b\) are parallel

Cross Product

• \(a \times b = \begin{bmatrix} a_yb_z - a_zb_y\\ a_zb_x - a_xb_z\\ a_xb_y - a_yb_x \end{bmatrix}\)

Notes:

• \(a \times b = 0\) iff \(a\) and \(b\) are parallel
• \(\vert a \times b \rvert = \vert a \rvert\vert b \rvert\) iff \(a\) and \(b\) are perpendicular

Parallel

• \(a \cdot b = \lvert a \rvert\lvert b \rvert\) iff \(a\) and \(b\) are parallel
• \((a \cdot b)^2 = \lvert a \rvert^2\lvert b \rvert^2\)
• \((a_xb_x + a_yb_y + a_zb_z)^2 = (a_x^2 + a_y^2 + a_z^2)(b_x^2 + b_y^2 + b_z^2)\)

• 11 multiplications
• 6 additions

• \(a \times b = 0\) iff \(a\) and \(b\) are parallel
• \(\begin{bmatrix} a_yb_z - a_zb_y\\ a_zb_x - a_xb_z\\ a_xb_y - a_yb_x \end{bmatrix} = 0\)

• 6 multiplications
• 3 subtractions

Perpendicular

• \(a \cdot b = 0\) iff \(a\) and \(b\) are perpendicular
• \(a_xb_x + a_yb_y + a_zb_z = 0\)

• 3 multiplications
• 2 addition

• \(\lvert a \times b \rvert = \lvert a \rvert\lvert b \rvert\) iff \(a\) and \(b\) are perpendicular
• \(\lvert a \times b \rvert^2 = \lvert a \rvert^2\lvert b \rvert^2\)
• \(\begin{aligned} &(a_yb_z - a_zb_y)^2 + (a_zb_x - a_xb_z)^2 + (a_xb_y - a_yb_x)^2 =\\ &(a_x^2 + a_y^2 + a_z^2)(b_x^2 + b_y^2 + b_z^2) \end{aligned}\)

• 16 multiplications
• 9 additions

Angle Between Two Vectors

• \(\theta = \cos^{-1}\left(\dfrac{a \cdot b}{\lvert a \rvert\lvert b \rvert}\right)\)
• \(\theta = \cos^{-1}\left(\dfrac{a_xb_x + a_yb_y + a_zb_z}{\sqrt{a_x^2 + a_y^2 + a_z^2}\sqrt{b_x^2 + b_y^2 + b_z^2}}\right)\)

• 1 inverse trig
• 2 square roots
• 1 division
• 10 multiplications
• 6 additions

• \(\theta = \sin^{-1}\left(\dfrac{\lvert a \times b\rvert}{\lvert a \rvert\lvert b \rvert}\right)\)
• \(\theta = \sin^{-1}\left(\dfrac{\sqrt{(a_yb_z - a_zb_y)^2 + (a_zb_x - a_xb_z)^2 + (a_xb_y - a_yb_x)^2}}{\sqrt{a_x^2 + a_y^2}\sqrt{b_x^2 + b_y^2}}\right)\)

• 1 inverse trig
• 3 square roots
• 1 division
• 14 multiplications
• 7 additions