Vector Algebra Practice Questions (Medium)

A vector is a quantity that has both magnitude and direction, and vector algebra provides the tools to perform calculations and solve problems involving vectors.

Question 1: Find the dot product of the vectors:.

Then calculate the angle θ between the two vectors.

The dot product of two vectors and is given by:

Subtitute the components of and :

= (1)(4) + (2)(1) + (3)(5) = 21

The angle θ between two vectors is given by:​

Magnitude of the Vectors

After Factorizing the denominator

Cos θ

Question 2: The vectors form two adjacent sides of a parallelogram. Find:

1. ,
2. The area of the parallelogram.

The formula for the cross product of two vectors is:

Subtitue the values:

Expand the determinant:

Compute each minor determinant:

1. For = (-4)(-1) - (2)(1) = 4 -2 = 2

2. For = (3)(−1) − (1)(1) = −3 − 1 = −4

3. For =(3)(2) − (1)(−4) = 6 + 4 = 10

Area of the Parallelogram

The area of the parallelogram is given by the magnitude of the cross product:

The magnitude of is:

Question 3: Given:

Find

Compute:

The formula for the cross product of two vectors is:

Subtitue the values:

Expand the determinant:

Compute each minor determinant:

1. For = (-4)(-1) - (2)(1) = 4 -2 = 2

2. For = (3)(−1) − (1)(1) = −3 − 1 = −4

3. For =(3)(2) − (1)(−4) = 6 + 4 = 10

Compute

The dot product of two vectors is:

Expand using the dot product formula:

= (−3)(−1) + (−3)(2)

Perform the calculations:

= 3 - 6 = -3

Question 4: Find the vector equation of a line passing through the points ((1, 2, 3) and (4, 5, 6). Express the line in the form , where λ is a scalar.

Given: Points (1, 2, 3) and (4, 5, 6)

The vector equation of a line is:

where is a position vector of one point, and is the direction vector.

Position vector of the first point:

Direction vector:

Equation of the line:

Simplify:

Question 5: Find the shortest distance between the skew lines:

Given:

Direction vectors of the lines:

Vector joining a point on Line 1 and Line 2:

Shortest distance formula:

Cross Product ​​:

Magnitude of Cross Product:

Dot Product :

= (2)(-3) + (-3)(-3) + (1)(3) = -6 + 9 + 3 = 6.

Shortest Distance:

Question 6: Find a vector parallel to the plane 2x + 3y − z = 52 that is perpendicular to the vector .

Given:
Plane 2x + 3y − z = 52, vector

A vector parallel to the plane is perpendicular to the plane's normal vector:

Let . For to be parallel to the plane:

Substitute:

2a + 3b − c = 0.

Assume a = 1, b = 1, and solve for c:

2(1) + 3(1) − c = 0  ⟹  c = 5

Parallel vector:

Question 7: Can a vector have direction angles α = 30^∘, β = 45^∘, and γ = 135^∘? If yes, find the direction cosines of the vector.

The direction cosines are given by: l = cos ⁡α, m = cos ⁡β, n =cos ⁡γ

Substitute the angles: l = cos⁡30^∘ = √3/2, m = cos⁡45^∘ = √2/2, n = cos⁡135^∘ = −(√2/2)

The condition for direction cosines is: l² + m² + n² = 1

Subtitute l = √3/2, m = √2/2, n = −(√2/2):

Add them together:

3/2 + 2/4 + 2/4 = 7/4

The sum of the squares of the direction cosines is:

l² + m² + n² ≠ 1

Thus, a vector cannot have direction angles α = 30^∘, β = 45^∘, and γ = 135^∘.

Question 8: A vector makes angles π/3​ with the x-axis and π/4​ with the y-axis. Find the angle it makes with the z-axis.

We know that if l, m and n are the direction cosines and  α, β and γ are the direction angles then,

l = cos ⁡α = cos π/3 = cos⁡ 60° = 1/2,

m = cos ⁡β = cos⁡ π/4 = Cos 45° = √2/2

We need to find γ, the angle with the z-axis.

Solve for n²

Using the property l² + m² + n² = 1, substitute l and m:

1/4 + 2/4 + n² = 1

3/4 + n² = 1

n² = 1 - 3/4

n² = 1/4

Since n = cos γ, we only take only positive values of n, assuming the vector points ibn the positive z-direction: n = 1/2

cos γ = n = 1/2

γ = cos^-1 (1/2) = π/3

The vector makes an angle of: γ = π/3 (60 degrees) with the z-axis

Unsolved Practice Questions

Question 1: Find the dot product of the vectors: , Then calculate the angle θ between the two vectors.

Question 2: The vectors: represent two adjacent sides of a parallelogram. Find:

• The cross product .
• The area of the parallelogram.

Question 3: Given:

find the scalar triple product

Question 4: Find the vector equation of a line passing through the points: P(2, −1, 3) and Q(4, 3, −2). Express the equation in the form:

Question 5: Find the shortest distance between the skew lines: and .

Question 6: Find a vector parallel to the plane x − 2y + 3z = 7 that is perpendicular to the vector

Question 7: Can a vector have direction angles α = 60^∘, β = 60^∘, γ = 60^∘? If yes, find the direction cosines.

Question 8: A vector makes an angle π/6 with the x-axis and π/4​ with the y-axis. Find the angle it makes with the z-axis.

Answer Key

1. Dot product = 0, angle between them = 90^∘.
2. Cross product = , Area of parallelogram = 5 √3​.
3. −14
4. 2√3
5. Cannot have these direction angles.
6. π/3​ or 60^∘.