Vector magnitude formula: |a| = √(x² + y² + z²)

The vector magnitude formula is |a| = √(x² + y² + z²) for a = xi + yj + zk; unit vector â = a/|a|.

\(\displaystyle |\vec a|=\sqrt{x^2+y^2+z^2}\) \(\displaystyle \hat a=\frac{\vec a}{|\vec a|}\)

What each letter means

\(\vec a=x\hat i+y\hat j+z\hat k\) the vector
\(|\vec a|\) its magnitude
\(\hat a\) the unit vector in its direction

When to use it

To find the length of a vector, a distance between position vectors, a speed from a velocity vector, or a unit vector v/|v|.

Worked example

Find the magnitude and the unit vector of \(\vec a=2\hat i-\hat j+2\hat k\).

\(|\vec a|=\sqrt{4+1+4}=\sqrt9=3\)

Answer: \(|\vec a|=3\), \(\hat a=\tfrac13(2\hat i-\hat j+2\hat k)\)

Common mistake

Adding the components before squaring. Square each component, add, then square root.

On your course

Course	In the exam
Class 12	Learn it: CBSE gives no formula booklet

From our own CBSE Maths formula sheets, in our words.

Practise and revise

Magnitude of a vector formula card: |a| = √(x² + y² + z²) for a = xi + yj + zk; unit vector â = a/|a|. CBSE Math Revision — Magnitude of a vector formula card. Download the card (PNG) to save or print it.

Questions

What is the vector magnitude formula?

The vector magnitude formula is |a| = √(x² + y² + z²) for a = xi + yj + zk; unit vector â = a/|a|. a = x i + y j + z k: the vector; | a|: its magnitude; a: the unit vector in its direction.

Is the magnitude of a vector given in the exam?

Class 12: learn it: CBSE gives no formula booklet. This comes from our own CBSE Maths formula sheets; your teacher has the official booklet.

How do I find a unit vector?

Divide the vector by its magnitude: v / |v|. The result has length 1 and points the same way.

Our own wording, examples and card, checked by CBSE Math Revision. Not produced or endorsed by CBSE or NCERT.