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Step 18: Multivariable calculus

Vectors, dot and cross products

Step 18 · Multivariable calculus topics

Work with vectors in three dimensions, use the dot product to find angles and projections, and use the cross product to find perpendicular vectors and areas.

Vectors in three dimensions

A vector in space has three components, ⟨a, b, c⟩: how far to move in x, y and z. Add vectors component by component, and scale one by multiplying every component. Its length comes from the Pythagorean theorem used twice.

∣v∣=a2+b2+c2|\mathbf{v}| = \sqrt{a^{2} + b^{2} + c^{2}}

Example: Length

Find the length of v = ⟨2, 3, 6⟩.

  1. 4 + 9 + 36 = 49.
  2. √49 = 7.

Answer7

Example: A unit vector

Find the unit vector in the direction of ⟨2, 3, 6⟩.

  1. Divide each component by the length, 7.

Answer⟨2/7, 3/7, 6/7⟩

Example: From point to point

Find the vector from P(1, 0, 2) to Q(4, 4, 2), and the distance PQ.

  1. Subtract P from Q: ⟨3, 4, 0⟩.
  2. Its length is √(9 + 16 + 0), which is 5.

Answer⟨3, 4, 0⟩, distance 5

The dot product

The dot product multiplies matching components and adds: u · v = u₁v₁ + u₂v₂ + u₃v₃. It also equals |u||v| cos θ, so it measures how much two vectors point the same way.

u⋅v=∣u∣∣v∣cos⁡θ\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta

Example: An angle

Find the angle between u = ⟨1, 2, 2⟩ and v = ⟨2, 1, −2⟩.

  1. u · v = 2 + 2 − 4 = 0.
  2. So cos θ = 0.

Answer90°: they are perpendicular

Example: Work

A force F = ⟨3, 0, 4⟩ newtons moves an object along d = ⟨5, 2, 0⟩ meters. How much work does it do?

  1. Work is F · d: 15 + 0 + 0.

Answer15 joules

Example: Projection

Find the projection of u = ⟨4, 2, 0⟩ onto v = ⟨1, 0, 0⟩.

  1. The projection is u⋅v∣v∣2 v\frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{v}|^{2}}\,\mathbf{v}.
  2. u · v = 4 and |v|² = 1.

Answer⟨4, 0, 0⟩: the part of u along the x-axis

The cross product

The cross product u × v is a vector at right angles to both u and v. Its length is |u||v| sin θ, the area of the parallelogram they span, and its direction follows the right-hand rule.

u×v=⟨u2v3−u3v2, u3v1−u1v3, u1v2−u2v1⟩\mathbf{u} \times \mathbf{v} = \langle u_2v_3 - u_3v_2,\ u_3v_1 - u_1v_3,\ u_1v_2 - u_2v_1 \rangle

Example: The basic directions

Find i × j, where i = ⟨1, 0, 0⟩ and j = ⟨0, 1, 0⟩.

  1. The components are 0·0 − 0·1, 0·0 − 1·0 and 1·1 − 0·0.
  2. That is ⟨0, 0, 1⟩.

Answerk = ⟨0, 0, 1⟩

Example: Area of a triangle

Find the area of the triangle with corners (0, 0, 0), (2, 0, 0) and (0, 3, 4).

  1. u = ⟨2, 0, 0⟩ and v = ⟨0, 3, 4⟩, so u × v = ⟨0, −8, 6⟩.
  2. Its length, 10, is the parallelogram's area.
  3. The triangle is half of the parallelogram.

Answer5

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Find the length of ⟨1, 2, 2⟩.

    Show answer

    3

  2. Find ⟨1, 3, −2⟩ · ⟨4, 0, 5⟩.

    Show answer

    −6

  3. Are ⟨2, −1, 3⟩ and ⟨1, 5, 1⟩ perpendicular?

    Show answer

    Yes: their dot product is 2 − 5 + 3, which is 0

  4. Find ⟨1, 0, 0⟩ × ⟨0, 0, 1⟩.

    Show answer

    ⟨0, −1, 0⟩

  5. Find the area of the parallelogram spanned by ⟨3, 0, 0⟩ and ⟨0, 4, 0⟩.

    Show answer

    12

  6. Find the angle between ⟨1, 0, 0⟩ and ⟨1, 1, 0⟩.

    Show answer

    45°

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