Angle Between Vectors
Vectors
Intro: Exact/decimal; returns degrees and radians.
Worked example
- $a=(2,0,−1), b=(1,3,4)$
- Dot product: $$\vec a\cdot\vec b = 2\cdot1 + 0\cdot3 + (-1)\cdot4 = 2 - 4 = -2.$$
- Norms: $$\|\vec a\|=\sqrt{2^2 + 0^2 + (-1)^2}=\sqrt{4+0+1}=\sqrt{5},\qquad \|\vec b\|=\sqrt{1^2+3^2+4^2}=\sqrt{1+9+16}=\sqrt{26}.$$
- Cosine of the angle: $$\cos\theta = \dfrac{\vec a\cdot\vec b}{\|\vec a\|\,\|\vec b\|} = \dfrac{-2}{\sqrt{5}\,\sqrt{26}} = \dfrac{-2}{\sqrt{130}}.$$
- Exact expression for the angle: $$\theta = \arccos\!\left(\dfrac{-2}{\sqrt{130}}\right).$$
- Decimal evaluation: $$\dfrac{-2}{\sqrt{130}}\approx -0.1754 \;\Rightarrow\; \theta\approx 100.1^{\circ}.$$
- Radians: $$\theta\approx 100.1^{\circ}\times\dfrac{\pi}{180}\approx 1.746\,\text{rad}.$$
- Sanity checks:
- • Since $\vec a\cdot\vec b<0$, the angle should be obtuse ($>90^{\circ}$) ✔️.
- • Bounds: $-1\le\cos\theta\le1$ holds because $-2/\sqrt{130}\in[-1,1]$ ✔️.
FAQs
Zero vector?
The angle is undefined if either vector is the zero vector (division by zero in the formula).
Units?
We report degrees and radians. You can request only one if you prefer.
Higher dimensions?
The same dot-product formula works in any dimension as long as the vectors have equal length.
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