Sample Problems

Question 1 : Find the value of λ if λ (a + b ) = c where a = 3i + 4j + k , b = i – j + 4k and c = 8i + 6j + 10k ?

Solution:

Given ,

a = 3i + 4j + k , b = i – j + 4k and c = 8i + 6j + 10k

Now λ (a + b ) = c can be written as ,

λa + λb = c

or

λ(3i + 4j + k ) + λ ( i – j + 4k) = 8i + 6j + 10k

comparing coefficients of i / j / k

4λ = 8 or

λ =2.

Question 2 : Find the additive Inverse of the vector A = 13i + 12j + 5k ?

Solution:

The additive inverse of a given vector is the negative of the same vector. So additive inverse of A is -13i – 12j – 5k

Question 3 : Calculate the work done by a Force of F = 2i – j + 4k , N that creates a displacement of D = 6i – 2j +k.

Solution:

Work done by force is given by the dot product of Force and displacement.

So work done W is ,

⇒ W = F.D = ( 2i – j + 4k ). (6i – 2j +k )

⇒ W = 12 + 2 + 4 = 18 J

So Work done by the given force equals to 18 Joules.

Question 4: If a and b are two vectors such that |a| = 2, |b| = 1/√2, and Find the angle between a and b , such that their cross product is a unit vector ?

Solution:

Given , cross product of a and b is a unit vector , |a × b| = 1.

Also ,

sinθ = |a × b| / (|a| |b|)

⇒ sinθ = 1/ (2 × 1/√2)

⇒ sinθ= 1/√2

⇒ θ= 45° , So angle between a and b is 45° .

Vectors are one of the most important concepts in mathematics. Vectors are quantities that have both magnitude and direction. A vector quantity is represented by an arrow above its head. Vectors help us understand the behaviour of directional quantities in 2D and 3D planes. Vectors are also used for determining the position and change of position of points.

Every vector follows a certain set of rules, known as the properties of vectors. It is highly important to know these properties to have a strong command of vector algebra. In this article, we will see the definition of a vector, the properties of vectors, and the properties of vector products.