Vector Magnitude
Vectors
Intro: Enter components separated by commas; we’ll compute $\sqrt{x^2+y^2+\cdots}$, simplify radicals, and provide a decimal approximation.
Worked example
- $‖(3, −4, 12)‖$
- Write the magnitude formula for $\vec{v}=(x,y,z)$: $$\|\vec{v}\|=\sqrt{x^2+y^2+z^2}.$$
- Substitute $x=3$, $y=-4$, $z=12$: $$\|\vec{v}\|=\sqrt{3^2+(-4)^2+12^2}.$$
- Square each component: $$3^2=9,\quad (-4)^2=16,\quad 12^2=144.$$
- Add inside the radical: $$9+16+144=169.$$
- Take the square root: $$\sqrt{169}=13.$$
- Therefore $$\boxed{\| (3,-4,12) \|=13}.$$
- Note: squaring removes any negative signs on components; the magnitude is always non-negative.
- ‖(6, −8)‖
- Magnitude in 2D: $$\|\vec{v}\|=\sqrt{x^2+y^2}=\sqrt{6^2+(-8)^2}=\sqrt{36+64}=\sqrt{100}=10.$$
- Alternatively, factor inside the radical first: $$\sqrt{4(9+16)}=\sqrt{4\cdot25}=2\cdot5=10.$$
- Answer: $$\boxed{10}.$$
FAQs
Fractions allowed?
Yes—fractions and decimals are squared the same way; we simplify exact radicals when possible and also show a decimal.
ND vectors?
Enter any number of components separated by commas. We compute $\sqrt{\sum v_i^2}$ for all entries.
Why choose Mathix?
- 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.
Vector norm
- Compute v_i^2 for each component
- Add all v_i^2 inside a single radical
- Take square root; simplify exactly if possible, otherwise provide decimal