3.1.3 Practice set: vectors

Vectors and their arithmetic

Vectors and their arithmetic (lesson 3.1.1)

0 of 7 correct

1.
$\mathbf{p} = (4, -1)$ and $\mathbf{q} = (-3, 2)$ . What is the second component of $\mathbf{p} + \mathbf{q}$ ?
2.
A vector runs from the point $(-2, 3)$ to the point $(4, -1)$ . What is its first component?
3.
What is the third component of $3\,(1, 0, -2) - (2, 5, 1)$ ?
4.
For which scalar $c$ is $(6, -4) + c\,(3, -2)$ the zero vector?
5.
$2\mathbf{x} + (1, 3) = (7, -5)$ . What is the second component of $\mathbf{x}$ ?
6.
$a\,(1, 1) + b\,(1, -1) = (5, 1)$ . What is $a$ ?
7.
In Python, how many items does $\texttt{2 * [3, 1, 4]}$ have?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

The dot product, length and angle

The dot product, length and angle (lesson 3.1.2)

0 of 7 correct

1.
Find $(3, -1, 2) \cdot (1, 4, 2)$
2.
Find the length of $(1, -2, 2, 4)$
3.
Normalise $(-5, 12)$ . What is the first component of its unit vector?
4.
For which $k$ are $(2, k, 1)$ and $(3, -1, 4)$ orthogonal?
5.
$\mathbf{u} \cdot \mathbf{v} = -6$ , $\lVert \mathbf{u} \rVert = 2$ and $\lVert \mathbf{v} \rVert = 6$ . What is the angle between them, in degrees?
6.
Find the signed length of the shadow of $(1, 7)$ on the line through $(3, 4)$
7.
$\lVert \mathbf{u} \rVert = 2$ , $\lVert \mathbf{v} \rVert = 5$ and $\mathbf{u} \cdot \mathbf{v} = -10$ . What is the angle between them, in degrees?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Unit vectors, angles and shadows

Write three functions, using lesson 3.1.2. Each takes NumPy arrays or plain lists; turn them into float arrays first with np.asarray(x, dtype=float).

unit(u) returns $\frac{1}{\lVert \mathbf{u} \rVert}\mathbf{u}$ . Raise ValueError for the zero vector.
angle_deg(u, v) returns the angle between $\mathbf{u}$ and $\mathbf{v}$ in degrees. Raise ValueError if the sizes differ or either vector is zero.
shadow(v, u) returns the signed length of the shadow of $\mathbf{v}$ on the line through $\mathbf{u}$ , $\frac{\mathbf{u} \cdot \mathbf{v}}{\lVert \mathbf{u} \rVert}$ . Raise ValueError if the sizes differ or $\mathbf{u}$ is zero.

The tests include parallel and opposite vectors whose computed cosine comes out just past 1 or $-1$ , so clip it into $[-1, 1]$ before np.arccos. They also pass lists, for which + and * would join and repeat rather than add and scale.

See the tests

import math

import numpy as np


def _raises_value_error(f, *args):
    try:
        f(*args)
    except ValueError:
        return True
    return False


def test_unit():
    """The unit vector of (3, 4) is (0.6, 0.8)"""
    expect_close(unit(np.array([3, 4])), np.array([0.6, 0.8]), what="unit((3, 4))")


def test_unit_length_one():
    """The unit vector of (2, −3, 6) has length 1 and points the same way"""
    e = unit(np.array([2, -3, 6]))
    expect_close(np.linalg.norm(e), 1.0, what="length of unit((2, −3, 6))")
    expect_close(e, np.array([2, -3, 6]) / 7, what="unit((2, −3, 6))")


def test_unit_zero():
    """The zero vector has no unit vector: ValueError"""
    assert _raises_value_error(unit, np.array([0.0, 0.0])), "expected ValueError for unit((0, 0))"


def test_angles():
    """(3, 1) makes 45° with (1, 2), 90° with (−1, 3) and 135° with (−2, 1)"""
    u = np.array([3, 1])
    expect_close(angle_deg(u, np.array([1, 2])), 45.0, what="angle between (3, 1) and (1, 2)")
    expect_close(angle_deg(u, np.array([-1, 3])), 90.0, what="angle between (3, 1) and (−1, 3)")
    expect_close(angle_deg(u, np.array([-2, 1])), 135.0, what="angle between (3, 1) and (−2, 1)")


def test_angle_r3():
    """In three dimensions, (1, 1, 0) and (0, 1, 1) make 60°"""
    expect_close(angle_deg(np.array([1, 1, 0]), np.array([0, 1, 1])), 60.0, what="angle between (1, 1, 0) and (0, 1, 1)")


def test_angle_lists():
    """Plain lists work too: [3, 1] and [1, 2] make 45°"""
    expect_close(angle_deg([3, 1], [1, 2]), 45.0, what="angle_deg([3, 1], [1, 2])")


def test_parallel():
    """Parallel vectors give 0° and opposite ones 180°, even when rounding pushes the cosine past ±1"""
    for given, c in [([3, 1], 0.1), ([3, 1], 1.3), ([1, 2, 3], 1.1), ([2, 3, 6], 0.3), ([1, 1, 1], 1.3)]:
        u = np.array(given, dtype=float)
        same = angle_deg(u, c * u)
        assert not math.isnan(same), f"the angle between {given} and {c} times it came out as nan: clip the cosine into [−1, 1]"
        expect_close(same, 0.0, what=f"angle between {given} and {c} times it", abs_tol=1e-6)
        opposite = angle_deg(u, -c * u)
        assert not math.isnan(opposite), f"the angle between {given} and −{c} times it came out as nan: clip the cosine into [−1, 1]"
        expect_close(opposite, 180.0, what=f"angle between {given} and −{c} times it", abs_tol=1e-6)


def test_angle_zero_vector():
    """A zero vector has no angle: ValueError"""
    assert _raises_value_error(angle_deg, np.array([0, 0]), np.array([1, 2])), "expected ValueError when u is (0, 0)"
    assert _raises_value_error(angle_deg, np.array([3, 1]), np.array([0, 0])), "expected ValueError when v is (0, 0)"


def test_angle_sizes():
    """Vectors of different sizes have no angle: ValueError"""
    assert _raises_value_error(angle_deg, np.array([1, 2]), np.array([1, 2, 3])), "expected ValueError for (1, 2) and (1, 2, 3)"


def test_shadow():
    """The shadow of (1, 2) on (3, 1) has length 5/√10, and of (−2, 1) it is −5/√10"""
    expect_close(shadow(np.array([1, 2]), np.array([3, 1])), 5 / math.sqrt(10), what="shadow of (1, 2) on (3, 1)")
    expect_close(shadow(np.array([-2, 1]), np.array([3, 1])), -5 / math.sqrt(10), what="shadow of (−2, 1) on (3, 1)")


def test_shadow_divides_by_u():
    """The shadow of (1, 7) on (3, 4) is 6.2: divide by the length of (3, 4)"""
    expect_close(shadow(np.array([1, 7]), np.array([3, 4])), 6.2, what="shadow of (1, 7) on (3, 4)")


def test_shadow_bad_inputs():
    """No line through the zero vector, and no shadow across sizes: ValueError"""
    assert _raises_value_error(shadow, np.array([1, 2]), np.array([0, 0])), "expected ValueError when u is (0, 0)"
    assert _raises_value_error(shadow, np.array([1, 2, 3]), np.array([3, 1])), "expected ValueError for sizes 3 and 2"

Vectors and their arithmetic

The dot product, length and angle

Coding exercise

Free further reading