Comprehensive assessment covering vectors in three dimensions, vector operations and magnitude, the scalar (dot) product and the angle between vectors, projections, the vector, parametric and Cartesian equations of a line, intersecting and skew lines, and vector proof.
The magnitude of $\mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$ is:
If $\mathbf{a} = (1, 2, 3)$ and $\mathbf{b} = (4, -1, 2)$, then $\mathbf{a} + \mathbf{b}$ equals:
For $\mathbf{a} = (1, 2, 3)$ and $\mathbf{b} = (4, -1, 2)$, the dot product $\mathbf{a} \cdot \mathbf{b}$ is:
Two non-zero vectors are perpendicular if and only if their dot product equals:
The angle $\theta$ between two non-zero vectors $\mathbf{a}$ and $\mathbf{b}$ satisfies:
Which pair of vectors is perpendicular?
A unit vector in the direction of $\mathbf{a} = (2, -1, 2)$ is:
The vector equation of the line through the point with position vector $\mathbf{a}$, parallel to $\mathbf{d}$, is:
On the line $\mathbf{r} = (1, 0, 2) + \lambda(2, 1, -1)$, the point corresponding to $\lambda = 2$ is:
Two lines in three dimensions that do not intersect and are not parallel are said to be:
The scalar projection of $\mathbf{a}$ onto $\mathbf{b}$ is:
If $\mathbf{a} \cdot \mathbf{b} = 0$ and neither vector is the zero vector, then the angle between them is:
The vectors $(2, 4, 6)$ and $(1, 2, 3)$ are:
For any vector $\mathbf{a}$, the dot product $\mathbf{a} \cdot \mathbf{a}$ equals:
The lines $\mathbf{r} = (0, 0, 0) + \lambda(1, 1, 1)$ and $\mathbf{r} = (1, 0, 0) + \mu(1, 1, 1)$ are:
1. Find the magnitude of $\mathbf{a} = 4\mathbf{i} - 3\mathbf{k}$. (1 mark)
2. Given $\mathbf{a} = (2, 1, -2)$ and $\mathbf{b} = (1, 3, 0)$, find $\mathbf{a} \cdot \mathbf{b}$. (1 mark)
3. Find the angle between $\mathbf{a} = (1, 0, 0)$ and $\mathbf{b} = (1, 1, 0)$, in degrees. (2 marks)
4. Find a unit vector in the direction of $\mathbf{v} = (3, -4, 0)$. (2 marks)
5. Determine whether $\mathbf{a} = (1, 2, -1)$ and $\mathbf{b} = (3, -1, 1)$ are perpendicular, justifying your answer. (2 marks)
6. Write the vector equation of the line through $A(1, 2, 3)$ parallel to $\mathbf{d} = (2, -1, 4)$. (2 marks)
7. Find the scalar projection of $\mathbf{a} = (3, 4, 0)$ onto $\mathbf{b} = (1, 0, 0)$. (2 marks)
8. Show that the points $A(1, 1, 1)$, $B(2, 3, 5)$ and $C(3, 5, 9)$ are collinear. (2 marks)
9. Find the value of $t$ for which $(t, 2, 3)$ and $(4, -2, 1)$ are perpendicular. (2 marks)
10. Find the point of intersection of the lines $\mathbf{r} = (1, 0, 0) + \lambda(1, 1, 0)$ and $\mathbf{r} = (0, 1, 0) + \mu(1, -1, 0)$. (3 marks)