Dot Product

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Dot Product: meaning, method and solved example (Precalculus) – Maths PDF

In short: The dot product collapses two vectors into one ordinary number by multiplying matching components and adding, a₁b₁ + a₂b₂, a value tied to the cosine of the gap between the arrows.

PrecalculusGeometryN-VM.B.5

What is Dot Product?

The dot product measures how much two vectors point the same way. It returns an ordinary number, not a vector: large and positive for arrows that agree, zero for arrows at a right angle, and negative for arrows that oppose each other.

Given \(\mathbf{a} = \langle a_1, a_2\rangle\) and \(\mathbf{b} = \langle b_1, b_2\rangle\), the dot product is \(\mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2\). Geometrically, \(\mathbf{a}\cdot\mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta\), with \(\theta\) the angle between them when they start at the same point.

How to spot it

  • A question asks whether two arrows are perpendicular, and a dot product of zero would settle it.
  • Work done by a force along a displacement is wanted, which multiplies the aligned parts.
  • A problem asks for the component of one vector in the direction of another.
  • A raised dot appears between two vectors written in angle brackets.

The method

  1. Multiply the first components together and the second components together.
  2. Add the two products to get one number.
  3. To find the angle, compute both lengths and use \(\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|}\).
  4. Read the sign: a zero dot product signals a right angle, positive an acute angle, negative an obtuse one.
a = ⟨5, 2⟩b = ⟨-1, 7⟩θ ≈ 76.3°a · b = 5(-1) + 2(7) = 9cos θ = 9 / (√29 · √50)
a · b = 9 gives cos θ ≈ 0.2364, so the angle is about 76.3°.

Dot Product: solved example

Problem

Take \(\mathbf{a} = \langle 5, 2\rangle\) and \(\mathbf{b} = \langle -1, 7\rangle\). Compute \(\mathbf{a}\cdot\mathbf{b}\), then the opening between the two arrows, rounded to a tenth of a degree.

Solution

Pair components: \(5(-1) + 2(7) = -5 + 14 = 9\). The lengths are \(\sqrt{29} \approx 5.385\) and \(\sqrt{50} \approx 7.071\).

Then \(\cos\theta = \frac{9}{5.385 \times 7.071} \approx 0.2364\), so \(\theta \approx 76.3^{\circ}\). The value is 9, with an angle of about 76.3°.

Try it

Question 1

Are \(\langle 4, -6\rangle\) and \(\langle 3, 2\rangle\) perpendicular?

Show the answer

\(4(3) + (-6)(2) = 12 – 12 = 0\), so yes.

Question 2

Find \(k\) so that \(\langle k, 5\rangle\) is perpendicular to \(\langle 10, -4\rangle\).

Show the answer

Set \(10k – 20 = 0\), so \(k = 2\).

Dot Product vs. Vector

Adding two vectors, or scaling one, gives another vector. This operation takes two vectors and gives back only a number, a scalar. Because of that, you cannot dot three vectors in a row and expect a vector; the first pairing already produces a plain number.

Vector

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