JEE Mains Maths · Vector Algebra

Magnitudes and Unit-Vector Identities

Working with vectors known only by their lengths and the angles between them: expanding the square of a sum, and handling combinations of unit vectors that include their cross product.

All Vector Algebra notes JEE Mains Maths strategy

Why this matters

Twenty-four PYQs, and none gives a single component. Everything is lengths, angles and dot products, so the only tool is squaring: |v|² = v · v, expanded term by term. Two ideas cover the page.

Concept 1 of 2: Expanding the square of a sum

A length is found through its square:

|\vec a+\vec b|^2=(\vec a+\vec b)\cdot(\vec a+\vec b)=|\vec a|^2+2\vec a\cdot\vec b+|\vec b|^2

. Any condition on lengths of combinations becomes, after squaring, an equation in

|\vec a|

|\vec b|

and

\vec a\cdot\vec b

. For a sum of three vectors, the cross terms are the three pairwise dot products. For a maximum of

p|\vec u|+q|\vec v|

when

|\vec u|^2+|\vec v|^2

is fixed, Cauchy–Schwarz gives

\sqrt{p^2+q^2}\sqrt{|\vec u|^2+|\vec v|^2}

Definition

$|\vec a\pm\vec b|^2=|\vec a|^2+|\vec b|^2\pm2\,\vec a\cdot\vec b$ .
$|\vec a+\vec b|^2+|\vec a-\vec b|^2=2(|\vec a|^2+|\vec b|^2)$ .
$|\vec a+\vec b+\vec c|^2=|\vec a|^2+|\vec b|^2+|\vec c|^2+2(\vec a\cdot\vec b+\vec b\cdot\vec c+\vec c\cdot\vec a)$ .
$|\vec a+\vec b|=|\vec a-\vec b|$ exactly when $\vec a\perp\vec b$ .

Square of a sum

|\vec a+\vec b|^2=|\vec a|^2+2\,\vec a\cdot\vec b+|\vec b|^2

Worked example

|\vec a|=3

|\vec b|=4

\vec a\cdot\vec b=6

. Find

|\vec a-\vec b|^2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 22 · Q66Moderate

Example 1 · Vector Algebra · Magnitudes and Unit-Vector Identities

Let

\overrightarrow{a}

and

\overrightarrow{b}

be two vectors such that

|2\overrightarrow{a}+ 3\overrightarrow{b}| = |3\overrightarrow{a}+\overrightarrow{b}|

and the angle between

\overrightarrow{a}

and

\overrightarrow{b}

60^{\circ}

. If

\frac{1}{8}\overrightarrow{a}

is a unit vector, then

|\overrightarrow{b}|

is equal to:

Square before you add

|\vec a+\vec b|

|\vec a|+|\vec b|

only when the vectors point the same way. Always square, expand, and take the root at the end.

Drill 11 more on expanding the square of a sum

Concept 2 of 2: Combinations with a cross-product term

\vec a\times\vec b

is perpendicular to both

\vec a

and

\vec b

, and for unit vectors its length is

\sin\theta

. So in a combination

\lambda\vec a+\mu\vec b+\nu(\vec a\times\vec b)

, dotting with

\vec a

\vec b

kills the cross term, and the square of its length splits into the in-plane part plus

\nu^2\sin^2\theta

Definition

$(\vec a\times\vec b)\cdot\vec a=(\vec a\times\vec b)\cdot\vec b=0$ .
Unit $\vec a,\vec b$ : $|\vec a\times\vec b|=\sin\theta$ , $\vec a\cdot\vec b=\cos\theta$ .
$|\lambda\vec a+\mu\vec b+\nu(\vec a\times\vec b)|^2=\lambda^2+\mu^2+2\lambda\mu\cos\theta+\nu^2\sin^2\theta$ .

Length with a cross term (unit vectors)

|\lambda\vec a+\mu\vec b+\nu(\vec a\times\vec b)|^2=\lambda^2+\mu^2+2\lambda\mu\cos\theta+\nu^2\sin^2\theta

Worked example

Unit vectors

\vec a,\vec b

are at

60^\circ

, and

\vec c=\vec a+\vec b+(\vec a\times\vec b)

. Find

|\vec c|^2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2021 · Paper 10 · Q73Moderate

Example 2 · Vector Algebra · Magnitudes and Unit-Vector Identities

Let

\overrightarrow{a}

and

\overrightarrow{b}

be two non-zero vectors perpendicular to each other and

|\overrightarrow{a}| = |\overrightarrow{b}|

. If

|\overrightarrow{a}\times\overrightarrow{b}| = |\overrightarrow{a}|

, then the angle between the vectors

(\overrightarrow{a}+\overrightarrow{b}+ (\overrightarrow{a}\times\overrightarrow{b}))

and

\overrightarrow{a}

is equal to:

Only for unit vectors

|\vec a\times\vec b|=\sin\theta

needs

|\vec a|=|\vec b|=1

. In general it is

|\vec a||\vec b|\sin\theta

Drill 11 more on combinations with a cross-product term

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Expanding the square of a sum
Square of a sum
$|\vec a+\vec b|^2=|\vec a|^2+2\,\vec a\cdot\vec b+|\vec b|^2$
Combinations with a cross-product term
Length with a cross term (unit vectors)
$|\lambda\vec a+\mu\vec b+\nu(\vec a\times\vec b)|^2=\lambda^2+\mu^2+2\lambda\mu\cos\theta+\nu^2\sin^2\theta$

Watch out for (2)

Square before you add
→ Expanding the square of a sum
Only for unit vectors
→ Combinations with a cross-product term

Test yourself on Vector Algebra

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Vector Algebra chapter test Drill all 24 questions