Angle Between Vectors (Trig)

Key idea

Using dot product and arccos.

In depth

cos θ = (a · b)/(|a||b|) gives the angle between two vectors. It arises from the law of cosines applied to the triangle formed by a, b, and a − b.

Worked example

a = (1, 0), b = (1, 1): cos θ = 1/√2, θ = 45°.

Formula

Real-world uses

• Measuring the angle between wind and aircraft direction
• Checking if two forces push in the same way

Common mistake

Forgetting to divide by both magnitudes.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Trigonometry

• Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
• Phase Shift – Horizontal translation of a wave.
• Vertical Shift and Midline – Moves the wave up or down.
• General Sinusoid – Combine amplitude, period, phase and shift.
• Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
• Pythagorean Identity – The key identity from the unit circle.