JEE Mains Maths · Vector Algebra

Triple Products and Coplanarity

The scalar triple product as a determinant, a volume and a test for coplanarity, and the vector triple product a × (b × c) expanded into dot products.

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Why this matters

Twenty-nine PYQs. The scalar triple product questions are coplanarity tests and volumes, one determinant each. The vector triple product questions look heavy but collapse to two terms once expanded. Two ideas cover the page.

Concept 1 of 2: Scalar triple product: volume and coplanarity

[\vec a\,\vec b\,\vec c]=\vec a\cdot(\vec b\times\vec c)

is the determinant of the three component rows, and its absolute value is the volume of the parallelepiped on the three edges. Zero volume means the three vectors lie in one plane. Four points are coplanar when the three edges from one of them are. Swapping two vectors changes the sign; a cyclic shift does not.

Definition

$[\vec a\,\vec b\,\vec c]=\begin{vmatrix}a_1&a_2&a_3\\b_1&b_2&b_3\\c_1&c_2&c_3\end{vmatrix}$ .
Volume of the parallelepiped $=|[\vec a\,\vec b\,\vec c]|$ ; tetrahedron $=\frac16$ of it.
Coplanar exactly when $[\vec a\,\vec b\,\vec c]=0$ ; points $A,B,C,D$ : $[\overrightarrow{AB}\,\overrightarrow{AC}\,\overrightarrow{AD}]=0$ .
$[\vec a+\vec b,\,\vec b+\vec c,\,\vec c+\vec a]=2[\vec a\,\vec b\,\vec c]$ .

Coplanarity

[\vec a\ \vec b\ \vec c]=\vec a\cdot(\vec b\times\vec c)=0

Worked example

Find the volume of the parallelepiped with edges

(1,0,0)

(1,1,0)

(1,1,1)

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2023 · 25 Jan 2023 · Q152Moderate

Example 1 · Vector Algebra · Triple Products and Coplanarity

If the four points, whose position vectors are

3\widehat{i}- 4\widehat{j}+ 2\widehat{k},\widehat{i}+ 2\widehat{j}-\widehat{k}, - 2\widehat{i}-\widehat{j}+ 3\widehat{k}

and

5\widehat{i}- 2\alpha\widehat{j}+ 4\widehat{k}

are coplanar, then

\alpha

is equal to

Points need edges, not positions

Four points are coplanar when the three edges from one of them are coplanar. Testing the four position vectors directly tests something else.

Drill 16 more on scalar triple product: volume and coplanarity

Concept 2 of 2: Vector triple product: a × (b × c)

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

lies in the plane of

\vec b

and

\vec c

, and equals

(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c

. Expand first, then use the given dot products: a long chain of crosses shrinks to two terms. When the result is compared with a combination of

\vec b

and

\vec c

that are not parallel, match the coefficients.

Definition

$\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\vec b-(\vec a\cdot\vec b)\vec c$ .
$(\vec a\times\vec b)\times\vec c=(\vec a\cdot\vec c)\vec b-(\vec b\cdot\vec c)\vec a$ .
Crossing with $\hat i$ : $(v_1,v_2,v_3)\times\hat i=(0,v_3,-v_2)$ .

BAC − CAB

\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\,\vec b-(\vec a\cdot\vec b)\,\vec c

Worked example

\vec a=\hat i

\vec b=\hat j

\vec c=(1,1,1)

. Find

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

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2022 · 29 July 2022 · Q70Moderate

Example 2 · Vector Algebra · Triple Products and Coplanarity

Let

\overrightarrow{a}= 3\widehat{i}+\widehat{j}

and

\overrightarrow{b}=\widehat{i}+ 2\widehat{j}+\widehat{k}

. Let

\overrightarrow{c}

be a vector satisfying

\overrightarrow{a}\times (\overrightarrow{b}\times\overrightarrow{c}) =\overrightarrow{b}+ \lambda\overrightarrow{c}

. If

\overrightarrow{b}

and

\overrightarrow{c}

are non-parallel, then the value of

\lambda

is:

Not associative

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

and

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

are different vectors. Check which pair is inside the bracket before expanding.

Drill 11 more on vector triple product: a × (b × c)

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)

Scalar triple product: volume and coplanarity
Coplanarity
$[\vec a\ \vec b\ \vec c]=\vec a\cdot(\vec b\times\vec c)=0$
Vector triple product: a × (b × c)
BAC − CAB
$\vec a\times(\vec b\times\vec c)=(\vec a\cdot\vec c)\,\vec b-(\vec a\cdot\vec b)\,\vec c$

Watch out for (2)

Points need edges, not positions
→ Scalar triple product: volume and coplanarity
Not associative
→ Vector triple product: a × (b × c)

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.

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