关于向量的更多内容

The standard basis gives vectors component notation

Linear combinations build new vectors from a few directions

Lines come from a point and a direction

Parametrization is a general tool for describing paths

Write the vector $(4, -3, 2)$ using the standard basis vectors $\mathbf{i}$, $\mathbf{j}$ and $\mathbf{k}$ of $\mathbb{R}^3$. Which expression is correct?

A line in space has parametric equations $x = 1 + 2t$, $y = 3 - t$, $z = 5 + 3t$. Which point lies on the line at parameter value $t = 1$?

A line passes through $P(1, 2, 1)$ and $Q(3, -1, 4)$. Which vector can serve as a direction vector for the line?

Let $\mathbf{u} = (1, 2, -1)$ and $\mathbf{v} = (0, 1, 3)$. Which vector equals the combination $2\mathbf{u} - \mathbf{v}$?

A delivery drone flies straight from $A(1, 1, 2)$ to $B(4, 3, 0)$. Write the point halfway along its path as three comma-separated numbers, like 1,2,3.

With $\mathbf{u} = (1, 1)$ and $\mathbf{v} = (1, -1)$, the combination $a\mathbf{u} + b\mathbf{v}$ equals $(6, 2)$. Give the weights in the order $a$ then $b$, comma-separated, like 3,2.