Each worksheet prints cleanly on A4 or Letter. Answer keys are on a separate page.

The Lumenary · Grades 9–12

Vectors and coordinate geometry

Name: ______________________

Date: ______________________

You will learn to

Add vectors and multiply them by scalars, and interpret both geometrically and in coordinates
Use the dot product to compute lengths, angles and projections, and to test perpendicularity
Prove the Cauchy–Schwarz and triangle inequalities, including when equality holds
Write vector and parametric equations of lines in the plane and in space, and of planes in space
Prove geometric facts, such as that the diagonals of a parallelogram bisect each other, using vectors

1.
Find the length of the vector $v = (2, -3, 6)$ .
2.
Compute $(1, 2, 3) \cdot (4, -5, 6)$ .
3.
For which number $k$ is $(2, k)$ perpendicular to $(3, -6)$ ?
4.
Find the angle, in degrees, between $u = (1, 1, 0)$ and $v = (1, 0, 1)$ .
5.
Let $u, v, w$ be vectors in $\R^3$ and $t$ a number. Which of these expressions are vectors? (Choose all that apply.)
- A. $u + t v$
- B. $(u \cdot v)\, w$
- C. $u \cdot v$
- D. $|u - v|$
6.
For non-zero vectors $u$ and $v$ , when is $|u + v| = |u| + |v|$ ?
- A.Exactly when $v = t u$ for some $t > 0$
- B.Exactly when $u \perp v$
- C.Exactly when $u$ and $v$ are parallel
- D.Always
7.
At which value of $t$ does the line $(1, 0, 0) + t\,(1, 2, 2)$ meet the plane $x + y + z = 6$ ?
8.
In the vector proof that the diagonals of a parallelogram $ABCD$ bisect each other, which equation expresses ' $ABCD$ is a parallelogram' (with vertices in this order)?
- A. $b - a = c - d$
- B. $b - a = d - c$
- C. $|b - a| = |c - d|$
- D. $(b - a) \cdot (d - a) = 0$
9.
In Mathlib, ⟪u, v⟫ is the dot product and ‖u‖ the length. Which statement is the Cauchy–Schwarz inequality?
- A.|⟪u, v⟫| ≤ ‖u‖ * ‖v‖
- B.‖u + v‖ ≤ ‖u‖ + ‖v‖
- C.⟪u, v⟫ = ‖u‖ * ‖v‖
- D.|⟪u, v⟫| ≤ ‖u‖ + ‖v‖
10.
Write projection(u, v) that returns the projection of $u$ onto a non-zero vector $v$ , $\dfrac{u\cdot v}{v \cdot v}\,v$ , as a list of Fractions (vectors are lists of integers of the same length, in any dimension). Raise ValueError if $v$ is the zero vector.
Computational — Practice

Answer key — Vectors and coordinate geometry

1.
$|v| = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7$ .
2.
$(1, 2, 3)\cdot(4, -5, 6) = 4 - 10 + 18 = 12$ . The answer is a number, not a vector.
3.
$(2, k) \cdot (3, -6) = 6 - 6k = 0$ gives $k = 1$ . Check: $(2, 1)\cdot(3, -6) = 6 - 6 = 0$ .
4.
$u \cdot v = 1$ and $|u| = |v| = \sqrt2$ , so $\cos\theta = \dfrac{1}{2}$ and $\theta = 60^\circ$ . (These are two face diagonals of a cube meeting at a corner; with a third diagonal they form an equilateral triangle, which explains the $60^\circ$ .)
5.
$u + tv$ and $(u \cdot v)\,w$ are vectors. $u \cdot v$ and $|u - v|$ are numbers. Keeping track of which is which avoids meaningless expressions like $u \cdot v \cdot w$ with two dots.
6.
Squaring, $|u + v| = |u| + |v|$ is equivalent to $u \cdot v = |u|\,|v|$ , i.e. $\cos\theta = 1$ , $\theta = 0$ . So $v$ is a positive multiple of $u$ . Parallel vectors pointing in opposite directions do not work.
7.
Substituting $(1 + t, 2t, 2t)$ gives $1 + 5t = 6$ , so $t = 1$ and the point is $(2, 2, 2)$ ; check $2 + 2 + 2 = 6$ .
8.
' $ABCD$ is a parallelogram' means $\overrightarrow{AB} = \overrightarrow{DC}$ , i.e. $b - a = c - d$ . Then $a + c = b + d$ , so the midpoints $\tfrac12(a + c)$ and $\tfrac12(b + d)$ of the diagonals coincide.
9.
Cauchy–Schwarz is |⟪u, v⟫| ≤ ‖u‖ * ‖v‖. Dividing by ‖u‖ * ‖v‖ (for non-zero vectors) gives |cos θ| ≤ 1, which is why the angle between vectors can be defined in any inner product space. The triangle inequality ‖u + v‖ ≤ ‖u‖ + ‖v‖ is a consequence.
10.
```
from fractions import Fraction

def projection(u, v):
    vv = sum(b * b for b in v)
    if vv == 0:
        raise ValueError('cannot project onto the zero vector')
    uv = sum(a * b for a, b in zip(u, v))
    c = Fraction(uv, vv)
    return [c * b for b in v]
```
The remainder $u - \operatorname{proj}_v u$ is perpendicular to $v$ because $(u - cv)\cdot v = u \cdot v - c\,(v\cdot v) = 0$ for $c = (u\cdot v)/(v \cdot v)$ . Exact fractions make the test $w \cdot v = 0$ exact, with no rounding. The formula works in any dimension: the four-dimensional test is no harder than the two-dimensional ones.