2D Vectors

In what follows are various notes and equations dealing with 2D vectors.

Magnitude

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

Distance Between Two Points

\(\lvert B - A \rvert = \sqrt{(B_x - A_x)^2 + (B_y - A_y)^2}\)

Dot Product

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

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

\(a \times b = a_xb_y - a_yb_x\)

Notes:

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

Parallel

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

• 8 multiplications
• 3 additions

• \(a \times b = 0\) iff \(a\) and \(b\) are parallel
• \(a_xb_y - a_yb_x = 0\)

• 2 multiplications
• 1 subtraction

Perpendicular

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

• 2 multiplications
• 1 addition

• \(a \times b = \lvert a \rvert\lvert b \rvert\) iff \(a\) and \(b\) are perpendicular
• \((a \times b)^2 = \lvert a \rvert^2\lvert b \rvert^2\)
• \((a_xb_y - a_yb_x)^2 = (a_x^2 + a_y^2)(b_x^2 + b_y^2)\)

• 8 multiplications
• 3 additions

Angle Between Two Vectors

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

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

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

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

Rotate

• \(R_{90^\circ}(x, y) = (-y, x)\)
• \(R_{180^\circ}(x, y) = (-x, -y)\)
• \(R_{270^\circ}(x, y) = (y, -x)\)
• \(s = \sin \theta\)
• \(c = \cos \theta\)
• \(R_\theta(x, y) = (xc - ys, xs + yc)\)