Algebra of dot products

Dot products are distributive: $$\mathbf{x} \cdot ( \mathbf{y} + \mathbf{z}) = \mathbf{x} \cdot \mathbf{y} + \mathbf{x} \cdot \mathbf{z}$$

As a result, the FOIL rule applies: $$(\mathbf{w} + \mathbf{x})\cdot(\mathbf{y} + \mathbf{z}) = \mathbf{w}\cdot \mathbf{y} + \mathbf{w}\cdot \mathbf{z} + \mathbf{x} \cdot \mathbf{y} + \mathbf{x} \cdot \mathbf{z}$$

Dot products are commutative: $$\mathbf{x}\cdot \mathbf{y} = \mathbf{y}\cdot \mathbf{x}$$

Practice problems

Derive the law of cosines using the dot product:
(a) Write $\text{CB}$ in terms of $\text{OB}$ and $\text{OC}$
(b) Write $\left | \text{CB} \right|^2$ in terms of $\left | \text{OB} \right |$, $\left | \text{OC} \right |$ and $\text{OB} \cdot \text{OC}$
(c) Show that $\left | \text{CB} \right|^2 = \left | \text{OB} \right|^2 + \left | \text{OC} \right |^2 - 2 \left | \text{OB} \right| \left | \text{OC} \right| \cos \theta$.
Solution
Show that the lines connecting any point on the semicircle of radius 1 to $(1,0)$ and $(-1,0)$ are perpendicular.
Solution
Use vectors and dot products to prove: if the diagonals of a rectangle are perpendicular, then the rectangle is a square.
Solution
Prove the parallelogram law: The sum of the squares of the lengths of both diagonals of a parallelogram equals the sum of the squares of the lengths of all four sides.
Solution
Prove for 2d vectors: $\bfa \cdot (\bfb + \bfc) = \bfa\cdot \bfb + \bfa \cdot \bfc.$
Solution
Does the vector projection of $\bfx$ along $\bfv$ depend on the length of $\bfv$? That is, if $\bfv$ is scaled by a scalar $c$, does the projection change?
Solution

Algebra of dot products

Related topics

Practice problems