Practice Questions on Vector Algebra (Hard)

Vector algebra is the branch of mathematics that deals with vector quantities that have both magnitude (size) and direction.

Question 1: Show that the points are the vertices of a right-angled triangle.

Let,

=>
=>
=>

The line segments are:

The magnitude of the sides are,

As we can see that

=> Thus, ABC is a right-angled triangle.

Question 2: The area (in sq. units) of the parallelogram whose diagonals are along the vectors is?

Diagonals of the parallogra is given by , then its area is calculated as:

Given,

Area of parallelogram

Question 3: If are vectors such that then = ?

Simplify using vector triple product identity:

Substitude back:

= 4² = 16

Question 4: Let

If the vector lies on the plane of and , then x equals?

Condition lies in the plane of

Expand the deteminant:

Compute each minor:

Subtitude back:

Simplify:

x = 1/2

Question 5: For any vector , the value of is equal to?

Let

Compute:

Magnitude squared:

Compute:

Magnitude Squared:

Sum of squares:

Simplify:

Using

Question 6: Let be three given vectors.

If is a vector such that and , then is equal to? [JEE Main 2024- April]

We are given:

It is stated that:

Now

-593λ - 67 = 0

λ = -(67/593)

= 569

Question 7: Let

If then is equal to? [JEE Main 2023- January]

Question 8: Let

Then is equal to? [JEE Main 2022- July]

Given:

Also,

Now,

= 5

Practice Questions

Question 1: The area ( in square units) of the paralleogram whose diagonals are along the vectors , is?

Question 2: If are vectors such that , then find .

Question 3: Let

If the vector lies in the plane of find x.

Question 4: For any vector , calculate the value of

Question 5: The values of a, for which the points P, Q, R with position vectors

respectively, are the vertices of a right-angled triangle with ∠PQR=π/2 are?

Question 6: Let

If the vector lies in the plane of , then x equals?

Question 7: If and the angle between and is π/4 then is equal to?

Question 8: If the vectors are such that

form a right-handed system, then is?

Answer Key

1. 33.96
2. 25
3. 3/2
4. -(5/2)
5. 1/2
6. 112.5