Ncert Solutions for Class 12
Vector Algebra
Vector Algebra
1. Write down a unit vector in XY-plane, making an angle of (30^°) with the positive direction of x-axis.
2. Find the scalar components and magnitude of the vector joining the points (P(x_1,y_1,z_1)) and (Q(x_2,y_2,z_2)).
3. A girl walks 4 km towards west, then she walks 3 km in a direction (30^°) east of north and stops. Determine the girl’s displacement from her initial point of departure.
4. If (vec{a}=vec{b}+vec{c}), then is it true that (left|vec{a}right|=left|vec{b}right|+left|vec{c}right|)? Justify your answer.
5. Find the value of x for which (xleft(hat{i}+hat{j}+hat{k}right)) is a unit vector.
6. Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
(vec{a}=2hat{i}+3hat{j}-hat{k}) and (vec{b}=hat{i}-2hat{j}+hat{k}).
7. If (vec{a}=hat{i}+hat{j}+hat{k}), (vec{b}=2hat{i}-hat{j}+3hat{k}) and (vec{c}=hat{i}-2hat{j}+hat{k}), find a unit vector parallel to the vector (2vec{a}-vec{b}+3vec{c}).
8. Show that the points A(1,–2,–8), B(5,0,–2) and C(11,3,7) are collinear and find the ratio in which B divides AC.
9. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (left(2vec{a}+vec{b}right)) and (left(vec{a}-3vec{b}right)) externally in the ratio 1:2. Also, show that P is the mid point of the line segment RQ.
10. The two adjacent sides of a parallelogram are (2hat{i}-4hat{j}+5hat{k}) and (hat{i}-2hat{j}-3hat{k}). Find the unit vector parallel to its diagonal. Also, find its area.
11. Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are (frac{1}{sqrt3},frac{1}{sqrt3},frac{1}{sqrt3}).
12. Let (vec{a}=hat{i}+4hat{j}+2hat{k}), (vec{b}=3hat{i}-2hat{j}+7hat{k}) and (vec{c}=2hat{i}-hat{j}+4hat{k}). Find a vector (vec{d}) which is perpendicular to both (vec{a}) and (vec{b}), and (vec{c}.vec{d}=15).
13. The scalar product of the vector (hat{i}+hat{j}+hat{k}) with a unit vector along the sum of vectors (2hat{i}+4hat{j}-5hat{k}) and (lambdahat{i}+2hat{j}+3hat{k}) is equal to one. Find the value of (lambda).
14. If (vec{a},vec{b},vec{c}) are mutually perpendicular vectors of equal magnitudes, show that the vector (vec{a}+vec{b}+vec{c}) is equally inclined to (vec{a},vec{b}) and (vec{c}).
15. Prove that (left(vec{a}+vec{b}right).left(vec{a}+vec{b}right)=left|vec{a}right|^2+left|vec{b}right|^2), if and only if (vec{a},vec{b}) are perpendicular, given (vec{a}neq0,vec{b}neq0).
Choose the correct answer in Exercises 16 to 19.
16. If θ is the angle between two vectors (vec{a}) and (vec{b}), then (vec{a}.vec{b}geq0) only when
(A) (0<theta<frac{pi}{2})
(B) (0lethetalefrac{pi}{2})
(C) (0<theta<pi)
(D) (0lethetalepi)
17. Let (vec{a}) and (vec{b}) be two unit vectors and θ is the angle between them. Then (vec{a}+vec{b}) is a unit vector if
(A) (theta=frac{pi}{4})
(B) (theta=frac{pi}{3})
(C) (theta=frac{pi}{2})
(D) (theta=frac{2pi}{3})
18. The value of (hat{i}.left(hat{j}timeshat{k}right)+hat{j}.left(hat{i}xhat{k}right)+hat{k}.left(hat{i}xhat{j}right)) is
(A) 0 (B) -1 (C) 1 (D) 3
19. If θ is the angle between any two vectors (vec{a} ) and (vec{b}), then (left|vec{a}.vec{b}right|=left|vec{a}timesvec{b}right|) when θ is equal to
(A) (0) (B) (frac{pi}{4}) (C) (frac{pi}{2}) (D) (pi)