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Vectors

lesson · about 3 minutes

Dot products, lengths, perpendicular vectors, cross products.

Pen and paper is fine · no calculator needed why?

Opens at level 35. You're level 1. You can read and practice here now.

the idea

A vector is a list of numbers, like (2, −1, 3): an arrow with a direction and a length.

The dot product multiplies matching entries and adds them, giving one number. It is 0 exactly when the two arrows are perpendicular (at right angles). The length comes from Pythagoras: square the entries, add, and take the square root. The cross product of two 3D vectors is a new vector at right angles to both, and each of its components is a small difference of two products.

techniques

Dot product, and perpendicular

Any dot product, or the k that makes two vectors perpendicular.

Multiply matching entries: first with first, second with second, and so on.
Add the products into one number.
Perpendicular: write the dot product with k in it, set it to 0, and solve for k.

worked example

Example: For what value of k are (2, k, 1) and (3, 2, −4) perpendicular?

First entries: 2 × 3 = 6.
Second: k × 2, or 2k. Third: 1 × (−4) = −4.
Add and set to 0: 6 + 2k − 4 = 0, so 2k = −2.
k = −1.

Answer: −1

Length: square, add, root

The length of any vector.

Square each entry. Negative entries give positive squares.
Add the squares, then take the square root.

worked example

Example: What is the length of the vector (4, −4, 7)?

Squares: 16, 16 and 49.
16 + 16 + 49 = 81.
The square root of 81 is 9.

Answer: 9

Cross product, one component

One component of a × b, where a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃).

x-component: a₂b₃ − a₃b₂.
y-component: a₃b₁ − a₁b₃.
z-component: a₁b₂ − a₂b₁.
Each skips its own position and uses the next two in the cycle 1, 2, 3, 1, 2.

worked example

Example: What is the y-component of (2, −1, 3) × (1, 4, −2)?

y-component: a₃b₁ − a₁b₃.
3 × 1 − 2 × (−2) = 3 + 4 = 7.

Answer: 7

watch out for

Stopping at the list of products in a dot product. Add them into one number.
Adding the entries for a length. Square them, add, then take the square root.
Subtracting in the wrong order in a cross product. The z-component starts with a₁b₂.

practice

Dot product

worked example

What is (2, −5, 6) · (6, 0, −3)?

Answer: -6

Multiply matching entries: 2 × 6, (−5) × 0, 6 × (−3).
Add the products: 12 + 0 − 18 = −6.

Length of a vector

worked example

What is the length of the vector (6, 7, 6)?

Answer: 11

Length = √(sum of the squares of the entries).
√(6² + 7² + 6²) = √(36 + 49 + 36) = √121 = 11.

Perpendicular vectors

worked example

For what value of k are (5, 5) and (−5, k) perpendicular?

Answer: 5

Perpendicular means the dot product is 0: 5 × (−5) + 5 × k = 0.
That is −25 + 5k = 0, so k = 5.

Cross product

worked example

What is the y-component of (1, 1, 2) × (4, −3, −4)?

Answer: 12

With a = (1, 1, 2) and b = (4, −3, −4), the y-component of a × b is a₃b₁ − a₁b₃.
2 × 4 − 1 × (−4) = 8 − (−4) = 12.