Important Questions About 12 Vectors

Important Questions & Answers for Vector Algebra

Q. No. 1: Represent graphically a displacement of 40 km, 30° east of north.

Solution:

Class 12 Chapter 10 Imp Ques. 1 figure 1

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

Q. No. 2: Find the unit vector in the direction of the sum of the vectors

$\begin{array}{l} \vec{�} = 2 \hat{�} - \hat{�} + 2 \hat{�} \\ \vec{�} = - \hat{�} + \hat{�} + 3 \hat{�} \end{array}$

Solution:

\begin{array}{l} Let \vec{�} be the sum of \vec{�} and \vec{�} . \end{array}

Class 12 Chapter 10 Imp Ques. 2 figure 1

The unit vector is:

Class 12 Chapter 10 Imp Ques. 2 figure 2

Q. No. 3: Find the vector joining the points P(2, 3, 0) and Q(– 1, – 2, – 4) directed from P to Q.

Solution:

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:

Class 12 Chapter 10 Imp Ques. 3 figure 1

\begin{array}{l} Q. No. 4: Find a vector in the direction of a vector 5 \hat{�} - \hat{�} + 2 \hat{�} which has a magnitude of 8 units. \end{array}

Solution:

\begin{array}{l} � � � \vec{�} = 5 \hat{�} - \hat{�} + 2 \hat{�} \end{array}

Class 12 Chapter 10 Imp Ques. 4 figure 1

\begin{array}{l} Hence, the vector in the direction of vector 5 \hat{�} - \hat{�} + 2 \hat{�} which has a magnitude of 8 units is given by \end{array}

Class 12 Chapter 10 Imp Ques. 4 figure 2

\begin{array}{l} Q. No. 5: Show that the vector \hat{�} + \hat{�} + \hat{�} is equally inclined to the axes OX, OY and OZ. \end{array}

Solution:

\begin{array}{l} � � � \vec{�} = \hat{�} + \hat{�} + \hat{�} \end{array}

Class 12 Chapter 10 Imp Ques. 5 figure 1

\begin{array}{l} Therefore, the direction cosines of \vec{�} � � � (\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}) . \end{array}

\begin{array}{l} Let �, �, � be the angles formed by \vec{�} with the positive directions of x, y, and z-axes. \end{array}

Then,

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.

Q. No. 6: Show that the points A, B and C with position vectors

$\begin{array}{l} \vec{�} = 3 \hat{�} - 4 \hat{�} - 4 \hat{�}, \\ \vec{�} = 2 \hat{�} - \hat{�} + \hat{�} \end{array}$

$\begin{array}{l} \vec{�} = \hat{�} - 3 \hat{�} - 5 \hat{�} \end{array}$
form the vertices of a right-angled triangle.

Solution:

Position vectors of points A, B and C are respectively given as below.

\begin{array}{l} \vec{�} = 3 \hat{�} - 4 \hat{�} - 4 \hat{�}, \\ \vec{�} = 2 \hat{�} - \hat{�} + \hat{�} \end{array}

\begin{array}{l} \vec{�} = \hat{�} - 3 \hat{�} - 5 \hat{�} \end{array}

Class 12 Chapter 10 Imp Ques. 6 figure 1

Therefore, ABC is a right-angled triangle.

\begin{array}{l} Q. No. 7: Find a vector \vec{�} of magnitude 3 \sqrt{2} units which makes an angle of � / 4, � / 2 with y and z-axes, respectively. \end{array}

Solution:

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:

Class 12 Chapter 10 Imp Ques. 7 figure 1

Q. No. 8: Evaluate the product.

$\begin{array}{l} (3 \vec{�} - 5 \vec{�}) . (2 \vec{�} + 7 \vec{�}) \end{array}$

Solution:

Class 12 Chapter 10 Imp Ques. 8 figure 1

Q. No. 9: Find all vectors of magnitude 10√3 that are perpendicular to the plane of

$\begin{array}{l} \hat{�} + 2 \hat{�} + \hat{�} \end{array}$

$\begin{array}{l} - \hat{�} + 3 \hat{�} + 4 \hat{�} \end{array}$

Solution:

\begin{array}{l} � � � \vec{�} = \hat{�} + 2 \hat{�} + \hat{�} \\ \vec{�} = - \hat{�} + 3 \hat{�} + 4 \hat{�} \end{array}

Class 12 Chapter 10 Imp Ques. 9 figure 1

\begin{array}{l} Hence, the unit vector perpendicular to the plane of \vec{�} and \vec{�} is: \end{array}

Class 12 Chapter 10 Imp Ques. 9 figure 2

\begin{array}{l} Therefore, the vectors of magnitude 10 \sqrt{3} that are perpendicular to the plane of \vec{�} and \vec{�} are: \end{array}

Class 12 Chapter 10 Imp Ques. 9 figure 3

Q. No. 10: Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

Solution:

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.

Class 12 Chapter 10 Imp Ques. 10 figure 1

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

Practice Questions For Class 12 Maths Chapter 10 Vector Algebra

Using vectors find the area of triangle PQR, whose vertices are P(1, 2, 3), Q(2, -1, 4) and C(4, 5, -1).
Use vector method to show that P,Q,R are Collinear – P(3, -5, 1), Q(-1, 0, 8) and C(7, -10, -6)
Find the angles between the lines whose direction ratios are 3, 2, -6 and 1, 2, 2. Find the angles of a triangle PQR whose vertices are P(-1, 3, 2), B(2, 3, 5) and C(3, 5, -2).
Prove that the points P(-2, 4, 7), Q (3, -6, -8) and R(1,-2,-2) are collinear.
Find the angle between the lines whose direction ratios are: 2, -3, 4 and 1, 2, 1.
Using vectors, find the value of k such that the points (k, – 10, 3), (1, –1, 3) and (3, 5, 3) are collinear.
$\begin{array}{l} If A, B, C, D are the points with position vectors \hat{�} + \hat{�} - \hat{�}, 2 \hat{�} - \hat{�} + 3 \hat{�}, 2 \hat{�} - 3 \hat{�}, 3 \hat{�} - 2 \hat{�} + \hat{�} respectively. Find the projection of \vec{� �} along \vec{� �} . \end{array}$
Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.
Show that:
Show that each of the given three vectors is a unit vector:

Class 12 Chapter 10 Imp Ques practice 10

Important Questions About 12 Vectors

Important Questions & Answers for Vector Algebra

Practice Questions For Class 12 Maths Chapter 10 Vector Algebra

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