doing math

courses › Linear Algebra

Vectors

level 35 course

Dot products, lengths, perpendicular vectors, cross products.

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Opens at level 35.

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Builds on: Systems of equations (not open yet) · Pythagoras & coordinates (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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.

  1. Multiply matching entries: first with first, second with second, and so on.
  2. Add the products into one number.
  3. 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?

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

Answer: −1

Length: square, add, root

The length of any vector.

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

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

  1. Squares: 16, 16 and 49.
  2. 16 + 16 + 49 = 81.
  3. 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).

  1. x-component: a₂b₃ − a₃b₂.
  2. y-component: a₃b₁ − a₁b₃.
  3. z-component: a₁b₂ − a₂b₁.
  4. 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)?

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

Answer: 7

Tips by skill

  • TipDot product: Multiply matching entries, then add the products into one number.
  • TipLength of a vector: Square each entry, add the squares, and take the square root.
  • TipPerpendicular vectors: Write the dot product with k in it, set it equal to 0, and solve for k.
  • TipCross product: x: a₂b₃ − a₃b₂. y: a₃b₁ − a₁b₃. z: a₁b₂ − a₂b₁. Keep the order of each subtraction.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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