Is Product of Vectors part of the EAPCET Mathematics syllabus?

Yes. Product of Vectors is taught in Intermediate first year · Paper IA Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.

How many Mathematics questions are there in EAPCET?

80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.

What mistakes do students make in Product of Vectors?

Treating the cross product as commutative; b⃗ × a⃗ = −(a⃗ × b⃗). Dividing by |a⃗| instead of |b⃗| for the projection of a⃗ on b⃗; you divide by the length of the vector you project onto. Forgetting the factor ½ when finding the area of a triangle from |AB⃗ × AC⃗|. Expanding a⃗ × (b⃗ × c⃗) with the wrong signs; it is (a⃗ · c⃗)b⃗ − (a⃗ · b⃗)c⃗, with the outer vector dotted into the far one first.

EAPCET Mathematics · Intermediate first year · Paper IA

Product of Vectors

Q: How should I prepare Product of Vectors for EAPCET?

Write the scalar triple product as a 3 × 3 determinant — it is faster and less error-prone. Remember the order in a × (b × c): the expansion is (a·c)b − (a·b)c.

Product of Vectors covers the scalar (dot) and vector (cross) products and their extensions, and uses them to find angles, projections, areas and volumes.

Subject: Mathematics
Taught in: Intermediate first year · Paper IA
Mathematics in the paper: 80 of 160 questions

Key formulas and results

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos\theta$ ; vectors are perpendicular when $\vec{a} \cdot \vec{b} = 0$
$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}| \sin\theta$ = area of the parallelogram on $\vec{a}$ and $\vec{b}$
Projection of $\vec{a}$ on $\vec{b}$ $= \dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$
Area of triangle $ABC = \tfrac{1}{2}\left|\overrightarrow{AB} \times \overrightarrow{AC}\right|$
$[\vec{a}\ \vec{b}\ \vec{c}]$ = volume of the parallelepiped; it is 0 exactly when the vectors are coplanar
$\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\,\vec{b} - (\vec{a} \cdot \vec{b})\,\vec{c}$
Lagrange identity: $|\vec{a} \times \vec{b}|^{2} + (\vec{a} \cdot \vec{b})^{2} = |\vec{a}|^{2}|\vec{b}|^{2}$
$|\vec{a} \pm \vec{b}|^{2} = |\vec{a}|^{2} + |\vec{b}|^{2} \pm 2\,\vec{a} \cdot \vec{b}$
Projection vector of $\vec{a}$ on $\vec{b}$ : $\left(\dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|^{2}}\right)\vec{b}$
Area of a quadrilateral with diagonals $\vec{d}_{1}$ and $\vec{d}_{2}$ : $\tfrac{1}{2}|\vec{d}_{1} \times \vec{d}_{2}|$
Volume of the tetrahedron with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ : $\tfrac{1}{6}\left|[\vec{a}\ \vec{b}\ \vec{c}]\right|$
$(\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c})\,\vec{b} - (\vec{b} \cdot \vec{c})\,\vec{a}$
Shortest distance between the parallel lines $\vec{r} = \vec{a}_{1} + \lambda\vec{b}$ and $\vec{r} = \vec{a}_{2} + \mu\vec{b}$ : $\dfrac{|(\vec{a}_{2} - \vec{a}_{1}) \times \vec{b}|}{|\vec{b}|}$
$[\vec{a}\ \vec{b}\ \vec{c}] = [\vec{b}\ \vec{c}\ \vec{a}] = [\vec{c}\ \vec{a}\ \vec{b}] = -[\vec{a}\ \vec{c}\ \vec{b}]$ , and $[\vec{a} + \vec{b},\ \vec{c},\ \vec{d}] = [\vec{a}\ \vec{c}\ \vec{d}] + [\vec{b}\ \vec{c}\ \vec{d}]$

What this chapter covers

Scalar product and the angle between vectors
Projection of one vector on another
Vector product and area
Scalar triple product and volume
Vector triple product

How it is usually asked

Find the angle between two vectors
Find a unit vector perpendicular to two given vectors
Compute an area or a volume
Simplify a vector triple product

Understanding Product of Vectors

The scalar (dot) product a⃗ · b⃗ = |a⃗||b⃗| cos θ is a number. In components it is a₁b₁ + a₂b₂ + a₃b₃, so it gives the angle between two vectors quickly, and it is zero exactly when they are perpendicular. Dividing a⃗ · b⃗ by |b⃗| gives the projection of a⃗ on b⃗, the length of the shadow a⃗ casts along b⃗.

The vector (cross) product a⃗ × b⃗ is a vector perpendicular to both a⃗ and b⃗, with magnitude |a⃗||b⃗| sin θ, which is the area of the parallelogram they form. It is computed as a 3 × 3 determinant with i, j, k in the first row, and it changes sign when the order is reversed. Half of |AB⃗ × AC⃗| is the area of triangle ABC.

Combining the two gives the scalar triple product [a⃗ b⃗ c⃗] = a⃗ · (b⃗ × c⃗), the determinant of the three component rows. Its absolute value is the volume of the parallelepiped on the three vectors, and it is zero exactly when they are coplanar. EAPCET questions use these products to find angles, perpendicular unit vectors, areas, volumes and unknown constants for coplanarity.

Worked example

EAPCET-style question

The vectors $\hat{i} + \hat{j} + \hat{k}$ , $2\hat{i} - \hat{j} + 3\hat{k}$ and $\hat{i} + \lambda\hat{j} + 4\hat{k}$ are coplanar. The value of $\lambda$ is

(1)−8
(2)8
(3)−4
(4)4

Solution

Coplanar vectors have scalar triple product 0, so the determinant with rows $(1, 1, 1)$ , $(2, -1, 3)$ , $(1, \lambda, 4)$ must vanish.
Expand along the first row: $1(-4 - 3\lambda) - 1(8 - 3) + 1(2\lambda + 1)$ .
This simplifies to $-4 - 3\lambda - 5 + 2\lambda + 1 = -8 - \lambda$ .
Setting $-8 - \lambda = 0$ gives $\lambda = -8$ .

Answer: −8

Shortcuts and results to remember

For any unit vectors $\vec{a}, \vec{b}, \vec{c}$ : $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} \ge -\tfrac{3}{2}$ , because $|\vec{a} + \vec{b} + \vec{c}|^{2} \ge 0$ .
If $\vec{d} \times \vec{b} = \vec{c} \times \vec{b}$ , then $(\vec{d} - \vec{c}) \times \vec{b} = \vec{0}$ , so write $\vec{d} = \vec{c} + \lambda\vec{b}$ and find $\lambda$ from the other condition.
A vector perpendicular to both $\vec{a}$ and $\vec{b}$ is $\lambda(\vec{a} \times \vec{b})$ ; a vector in the plane of $\vec{b}$ and $\vec{c}$ is $\lambda\vec{b} + \mu\vec{c}$ .
The angle between $\vec{a}$ and $\vec{b}$ is obtuse exactly when $\vec{a} \cdot \vec{b} < 0$ (excluding the antiparallel case).

Common mistakes to avoid

Treating the cross product as commutative; b⃗ × a⃗ = −(a⃗ × b⃗).
Dividing by |a⃗| instead of |b⃗| for the projection of a⃗ on b⃗; you divide by the length of the vector you project onto.
Forgetting the factor ½ when finding the area of a triangle from |AB⃗ × AC⃗|.
Expanding a⃗ × (b⃗ × c⃗) with the wrong signs; it is (a⃗ · c⃗)b⃗ − (a⃗ · b⃗)c⃗, with the outer vector dotted into the far one first.

How to prepare it

Write the scalar triple product as a 3 × 3 determinant — it is faster and less error-prone.

Remember the order in a × (b × c): the expansion is (a·c)b − (a·b)c.

Once the formulas feel familiar, test them the way the exam does: practise Product of Vectors questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise Product of Vectors chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Product of Vectors part of the EAPCET Mathematics syllabus?: Yes. Product of Vectors is taught in Intermediate first year · Paper IA Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Product of Vectors for EAPCET?: Write the scalar triple product as a 3 × 3 determinant — it is faster and less error-prone. Remember the order in a × (b × c): the expansion is (a·c)b − (a·b)c.
How many Mathematics questions are there in EAPCET?: 80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in Product of Vectors?: Treating the cross product as commutative; b⃗ × a⃗ = −(a⃗ × b⃗). Dividing by |a⃗| instead of |b⃗| for the projection of a⃗ on b⃗; you divide by the length of the vector you project onto. Forgetting the factor ½ when finding the area of a triangle from |AB⃗ × AC⃗|. Expanding a⃗ × (b⃗ × c⃗) with the wrong signs; it is (a⃗ · c⃗)b⃗ − (a⃗ · b⃗)c⃗, with the outer vector dotted into the far one first.