Vector Algebra

Vectors and Their Types

Understand

A vector is a directed line segment \(\overrightarrow{AB}\) with initial point \(A\) and terminal point \(B\). Its length \(|\overrightarrow{AB}|\) is the magnitude.

Type	Meaning
Zero vector \(\vec0\)	Magnitude 0, direction undefined
Unit vector	Magnitude 1
Co-initial vectors	Same initial point
Collinear (parallel) vectors	Parallel to the same line
Equal vectors	Same magnitude and direction
Negative of \(\vec a\)	Same magnitude, opposite direction

The position vector of \(P(x,y,z)\) is \(\overrightarrow{OP}=x\hat i+y\hat j+z\hat k\), with magnitude \(\sqrt{x^2+y^2+z^2}\).

Direction cosines. If \(\vec r\) makes angles \(\alpha,\beta,\gamma\) with the axes, then \(l=\cos\alpha=\frac{x}{r}\), \(m=\cos\beta=\frac yr\), \(n=\cos\gamma=\frac zr\), and \(l^2+m^2+n^2=1\).

Key Concepts

Equal vectors: same magnitude and direction (position does not matter).
Unit vector in the direction of a: a / |a|.
Direction cosines satisfy l² + m² + n² = 1.

Formula Bank

Magnitude and unit vector

\[|\vec a|=\sqrt{a_1^2+a_2^2+a_3^2},\qquad \hat a=\frac{\vec a}{|\vec a|}\]

Key Points

Direction cosines

For \(\vec r=x\hat i+y\hat j+z\hat k\): \(l=\frac xr,\ m=\frac yr,\ n=\frac zr\) and \(l^2+m^2+n^2=1\).

Practice & Topic Test

Topic Practice

A fresh set of questions from this topic. Change answers freely – solutions appear after you submit.

Start Practice

Topic Test

Timed: up to 10 questions in 15 minutes.

Start Test

Topic Summary

Know the types; compute magnitudes and unit vectors from components.

Ready to practise this topic?

Create a free account to take practice sessions and tests, see detailed explanations and track your progress.

Create Student Account