Vectors

Need to know

What you need to know

• The magnitude of the vector product is given by $|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin\theta$, where $\theta$ is the angle between the vectors. This value represents the area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$.
• The vector product is anti-commutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$. The order matters!
• If $\mathbf{a}$ and $\mathbf{b}$ are parallel, then $\mathbf{a} \times \mathbf{b} = \mathbf{0}$ (the zero vector), since $\sin(0) = \sin(\pi) = 0$.
• The area of a triangle with adjacent sides represented by vectors $\mathbf{a}$ and $\mathbf{b}$ is $\frac{1}{2}|\mathbf{a} \times \mathbf{b}|$.

Explanation

Building with Vectors

1. Define the vectors involved: direction vectors for lines, position vectors for points.
2. Calculate the vector (cross) product of the direction vectors to find a common perpendicular vector, $\mathbf{n}$.
3. Find a vector connecting a point on the first object (e.g., line 1) to a point on the second (e.g., line 2).
4. Use the dot product to project the connecting vector onto the perpendicular vector $\mathbf{n}$ to find the required shortest distance.