Dot Product
Vectors
Intro: Enter two same-length vectors; we compute $\vec{a}\cdot\vec{b}$ and, when possible, the angle $\theta$ via $\cos\theta=\dfrac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|\,\|\vec{b}\|}$.
Worked example
- a=(2, −1, 4), b=(−3, 5, 0)
- Dot product (component-wise multiply & add): $$\vec{a}\cdot\vec{b}=2\cdot(-3)+(-1)\cdot 5 + 4\cdot 0=-6-5+0=-11.$$
- Magnitudes: $$\|\vec{a}\|=\sqrt{2^2+(-1)^2+4^2}=\sqrt{4+1+16}=\sqrt{21},\qquad \|\vec{b}\|=\sqrt{(-3)^2+5^2+0^2}=\sqrt{9+25+0}=\sqrt{34}.$$
- Angle formula: $$\cos\theta=\frac{\vec{a}\cdot\vec{b}}{\|\vec{a}\|\,\|\vec{b}\|}=\frac{-11}{\sqrt{21}\,\sqrt{34}}=\frac{-11}{\sqrt{714}}\approx -0.4117.$$
- Therefore $$\theta=\arccos\!\left(\frac{-11}{\sqrt{714}}\right)\approx \arccos(-0.4117)\approx 114.31^{\circ}.$$
- Summary: $$\boxed{\vec{a}\cdot\vec{b}=-11},\quad \boxed{\|\vec{a}\|=\sqrt{21}},\quad \boxed{\|\vec{b}\|=\sqrt{34}},\quad \boxed{\theta\approx114.31^{\circ}}.$$
- Checks (optional):
- • A negative dot product implies an obtuse angle ($\theta>90^{\circ}$) ✔️.
- • If either vector were zero, the angle would be undefined (division by zero in $\cos\theta$).
FAQs
Different dimensions?
Vectors must have the same length $n$; we compute $\sum_{i=1}^{n} a_i b_i$.
Orthogonal test?
If $\vec{a}\cdot\vec{b}=0$ and both vectors are nonzero, the angle is $90^{\circ}$ (orthogonal).
Units for the angle?
We report degrees by default; you can request radians.
Why choose MathGPT?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
How this calculator works
- Type or paste your function (LaTeX like
\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
- Review each step to understand which rule was applied and why.
- Practice with similar problems to lock in the method.