投影、方向角与功

Projection is the shadow of one vector on another

Every vector splits into a part along b and a part across it

Direction angles describe a vector's tilt against the axes

Work multiplies force by displacement in a shared direction

Let $\mathbf{a} = (2, 3, 1)$ and $\mathbf{b} = (0, 0, 2)$. What is the vector projection of $\mathbf{a}$ onto $\mathbf{b}$?

Let $\mathbf{a} = (1, 2, 2)$ and $\mathbf{b} = (3, -4, 0)$. What is the scalar projection of $\mathbf{a}$ onto $\mathbf{b}$?

Let $\mathbf{a} = (3, 4, 12)$. What is $\cos\gamma$, the cosine of the angle between $\mathbf{a}$ and the positive $z$-axis?

A constant force $\mathbf{F} = (10, 5)$ acts on a wagon, with coordinates measured along and across a straight path, and the wagon moves along $\mathbf{d} = (3, 4)$. How much work does the force do?

The vector $\mathbf{a} = (3, 4)$ is projected onto the $x$-axis. Write the perpendicular remainder, the vector $\mathbf{a} - \text{proj}\,\mathbf{a}$, as two comma-separated numbers, like 1,2.

A vector has direction cosines $\cos\alpha = \tfrac{2}{3}$ and $\cos\beta = \tfrac{2}{3}$. Using the identity that the three squared direction cosines sum to $1$, give $\cos^2\gamma$ as a fraction like 1/9.