Important Questions: Vectors | Mathematics (Maths) Class 12 - JEE PDF Download

Q1: Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
Ans: Vertices of a triangle ABC are A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).
Let AB and BC be the adjacent sides of triangle ABC.

Hence, the area of triangle ABC is √61/2 sq.units.

Q2: Evaluate the product.

Ans:


Q3: Show that the points A, B and C with position vectors

form the vertices of a right-angled triangle.
Ans: Position vectors of points A, B and C are respectively given as below.


Therefore, ABC is a right-angled triangle.

Q4: Find a vector in the direction of a vectorwhich has a magnitude of 8 units.
Ans: Let


Q5: Find the unit vector in the direction of the sum of the vectors

Ans: Let  be the sum of

The unit vector is:


Q6: Find all vectors of magnitude 10√3 that are perpendicular to the plane of

Ans:

Hence, the unit vector perpendicular to the plane ofis:

Therefore, the vectors of magnitude 10√3 that are perpendicular to the plane of  are:


Q7: Find a vector  of magnitude 3√2 units which makes an angle of π/4, π/2 with y and z-axes, respectively.
Ans: From the given,
m = cos π/4 = 1/√2
n = cos π/2 = 0
Therefore, l² + m² + n² = 1
l² + (½) + 0 = 1
l² = 1 – ½
l = ±1/√2
Hence, the required vector is:


Q8: Show that the vectoris equally inclined to the axes OX, OY and OZ.
Ans:

Therefore, the direction cosines of

Let α, β,γ be the angles formed by  the positive directions of x, y, and z-axes.
cos α = 1/√3, cos β = 1/√3 cos γ = 1/√3
Hence, the given vector is equally inclined to axes OX, OY and OZ.

Q9: Find the vector joining the points P(2, 3, 0) and Q(– 1, – 2, – 4) directed from P to Q.
Ans: Since the vector is to be directed from P to Q, clearly P is the initial point and Q is the terminal point.
P(2, 3, 0) = (x₁, y₁, z₁)
Q(-1, -2, -4) = (x₂, y₂, z₂)
Vector joining the points P and Q is:


Q10: Represent graphically a displacement of 40 km, 30° east of north.
Ans:

Hence, the vector OP represents the displacements of 40 km, 30° east of north.