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Cross, Or Vector, Product Of Vectors

Class 12th Mathematics RS Aggarwal Solution
Exercise 24
  1. Find ( vector {a} x vec{b} ) and | vector {a} x vec{b}| , when vector {a} = {i} -…
  2. Find ( vector {a} x vec{b} ) and | vector {a} x vec{b}| , when vector {a} = 2 {i}…
  3. Find ( vector {a} x vec{b} ) and | vector {a} x vec{b}| , when vector {a} = {i}-7…
  4. Find ( vector {a} x vec{b} ) and | vector {a} x vec{b}| , when vector {a} = 4 {i}…
  5. Find ( vector {a} x vec{b} ) and | vector {a} x vec{b}| , when vector {a} = 3…
  6. Find λ if ( 2 {i}+6 hat{j}+14 hat{k} ) x ( hat{i} - lambda hat{j}+7 hat{k}…
  7. If vector {a} = ( - 3 {i}+4 hat{j}-7 hat{k} ) and vector {b} = ( 6…
  8. Find the value of:i. ( {i} x hat{j} ) c. hat{k} + hat{i} hat{j} ii. (…
  9. Find the unit vectors perpendicular to both vector {a} and vector {b} when…
  10. Find the unit vectors perpendicular to both vector {a} and vector {b} when…
  11. Find the unit vectors perpendicular to both vector {a} and vector {b} when…
  12. Find the unit vectors perpendicular to both vector {a} and vector {b} when…
  13. Find the unit vectors perpendicular to the plane of the vectors vector {a} = 2…
  14. Find a vector of magnitude 6 which is perpendicular to both the vectors vector…
  15. Find a vector of magnitude 5 units, perpendicular to each of the vectors (…
  16. Find an angle between two vectors vector {a} and vector {b} with magnitudes…
  17. If vector {a} = ( {i} - hat{j} ) , vector {b} = ( 3 {j} - hat{k} ) and…
  18. If vector {a} = ( 4 {i}+5 hat{j} - hat{k} ) , vector {b} = ( {i}-4 hat{j}…
  19. Prove that | vector {a} x vec{b}| = ( vec{a} c. vec{b} ) tantheta , where…
  20. Write the value of p for which vector {a} = ( 3 {i}+2 hat{j}+9 hat{k} )…
  21. Verify that vector {a} x ( vec{b} + vec{c} ) = ( vec{a} + vec{b} ) + (…
  22. Verify that vector {a} x ( vec{b} + vec{c} ) = ( vec{a} + vec{b} ) + (…
  23. vector {a} = {i}+2 hat{j}+3 hat{k} and vector {b} = - 3 {i}-2 hat{j} +…
  24. vector {a} = ( 3 {i} + hat{j}+4 hat{k} ) and vector {b} = ( {i} - hat{j}…
  25. vector {a} = 2 {i} + hat{j}+3 hat{k} and vector {b} = {i} - hat{j} Find…
  26. vector {a} = 2 {i} and vector {b} = 3 {j} Find the area of the…
  27. and vector {d}_{2} = {i}-3 hat{j}+4 hat{k} Find the area of the…
  28. vector {d}_{1} = 2 {i} - hat{j} + hat{k} and vector {d}_{2} = 3 {i}+4…
  29. vector {d}_{1} = {i}-3 hat{j}+2 hat{k} and vector {d}_{2} = - {i}+2…
  30. vector {a} = - 2 {i}-5 hat{k} and vector {b} = {i}-2 hat{j} - hat{k} Find…
  31. vector {a} = 3 {i}+4 hat{j} and vector {b} = - 5 {i}+7 hat{j} . Find the…
  32. A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5) Using vectors, find the area of ΔABC…
  33. A(1, 2, 3), B(2, −1, 4) and C(4, 5, Δ1) ((considering Δ1 as 1 )) Using…
  34. A(3, −1, 2), B(1, −1, −3) and C(4, −3, 1) Using vectors, find the area of…
  35. A(1, −1, 2), B(2, 1, −1) and C(3, −1, 2). Using vectors, find the area of…
  36. A(3, −5, 1), B(−1, 0, 8) and C(7, −10, −6) Using vector method, show that the…
  37. A(6, −7, −1), B(2, −3, 1) and C(4, −5, 0). Using vector method, show that the…
  38. Show that the point A, B, C with position vectors ( 3 {i}-2 hat{j}+4 hat{k}…
  39. Show that the points having position vectors vector {a} , vec{b} , ( vec{c} =…
  40. Show that the points having position vector ( - 2 vector {a}+3 vec{b}+5…
  41. Find a unit vector perpendicular to the plane ABC, where the points A, B, C,…
  42. If vector {a} = ( {i}+2 hat{j}+3 hat{k} ) and vector {b} = ( {i}-3…
  43. If | vector {a}| = 2 , and | vector {a} x vec{b}| = 8 , find vector {a}…
  44. If | vector {a}| = 2 , |b| = 7 and ( vector {a} x vec{b} ) = ( 3…
Exercise 24
Question 1.

Find and , when

and


Answer:


Here,


We


have




⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


and



Question 2.

Find and , when

and


Answer:


Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒



Question 3.

Find and , when

and


Answer:


Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒



Question 4.

Find and , when

and


Answer:


Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒



Question 5.

Find and , when

and


Answer:


Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒



Question 6.

Find λ if .
Answer:


Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒


⇒



Question 7.

If and , find .

Verify that (i) and are perpendicular to each other

and (ii) and are perpendicular to each other.
Answer:


Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


If and are perpendicular to each other then,


⇒


i.e.,



And in the similar way, we have,



Hence proved.
Question 8.

Find the value of:

i. ii. iii.


Answer:

i.


The value of is, …


⇒


ii.


The value of is, … …


⇒


iii.


The value of is, … …


⇒



Question 9.

Find the unit vectors perpendicular to both and when

and


Answer:

Let be the vector which is perpendicular to then we have,


…where k is a scalor


Thus, we have r is a unit vector,


So,


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒


⇒



Question 10.

Find the unit vectors perpendicular to both and when

and


Answer:

Let be the vector which is perpendicular to then we have,


…where k is a scalar


Thus, we have r is a unit vector,


So,


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒


⇒



Question 11.

Find the unit vectors perpendicular to both and when

and


Answer:

Let be the vector which is perpendicular to then we have,


…where k is a scalar


Thus, we have r is a unit vector,


So,


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒


⇒



Question 12.

Find the unit vectors perpendicular to both and when

and


Answer:

Let be the vector which is perpendicular to then we have,


…where k is a scalar


Thus, we have r is a unit vector,


So,


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒


⇒



Question 13.

Find the unit vectors perpendicular to the plane of the vectors

and


Answer:

Let be the vector which is perpendicular to then we have,


…where k is a scalar


Thus, we have r is a unit vector,


So,


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒




Question 14.

Find a vector of magnitude 6 which is perpendicular to both the vectors

and .
Answer:

Let be the vector which is perpendicular to then we have,


…where k is a scalar


Thus, we have r is vector of magnitude 6,


So,


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒



Here, as r is of magnitude 6 thus,


k = 6,


Thus,



Question 15.

Find a vector of magnitude 5 units, perpendicular to each of the vectors

and, where and


Answer:



Let be the vector which is perpendicular to then we have,


…where k is a scalar


Thus, we have r is vector of magnitude 5,


So,


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒



Here, as r is of magnitude 5 thus,


k = 5,


Thus,



Question 16.

Find an angle between two vectors and with magnitudes 1 and 2 respectively and .
Answer:

We are given that and .


And,


So we have,


|| =


⇒


⇒


⇒



Question 17.

If , and , find a vector which is perpendicular to both and and for which .
Answer:

Given that


Let be the vector which is perpendicular to then we have,


…where k is a scalar


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒



Given that



⇒


⇒


⇒



Question 18.

If , and , find a vector which is perpendicular to both and and for which .
Answer:

Given that


Let be the vector which is perpendicular to then we have,


…where k is a scalar


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒



Given that



⇒


⇒




Question 19.

Prove that , where θ is the angle between and .
Answer:

We know that |


And |


So,



Hence, proved.
Question 20.

Write the value of p for which and are parallel vectors.
Answer:

As the vectors are parallel vectors so,


Thus,


We have,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ Thus, p = .
Question 21.

Verify that , when

, and


Answer:

To verify


We need to prove L.H.S = R.H.S


L.H.S we have,


Given,




Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


RHS is



⇒


Thus, LHS = RHS.
Question 22.

Verify that , when

, and .
Answer:

To verify


We need to prove L.H.S = R.H.S


L.H.S we have,


Given, ,




Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


RHS is



⇒


Thus, LHS = RHS.
Question 23.

Find the area of the parallelogram whose adjacent sides are represented by the vectors:

and


Answer:

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 24.

Find the area of the parallelogram whose adjacent sides are represented by the vectors:

and


Answer:

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 25.

Find the area of the parallelogram whose adjacent sides are represented by the vectors:

and


Answer:

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 26.

Find the area of the parallelogram whose adjacent sides are represented by the vectors:

and


Answer:

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 27.

Find the area of the parallelogram whose diagonal are represented by the vectors

and


Answer:

The diagonals are


Thus,


The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒


⇒ sq units
Question 28.

Find the area of the parallelogram whose diagonal are represented by the vectors

and


Answer:

The diagonals are


Thus,


The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.


Area =



Here,


We


have
,


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒


⇒ sq units
Question 29.

Find the area of the parallelogram whose diagonal are represented by the vectors

and .
Answer:

The diagonals are


Thus,


The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒


⇒ sq units
Question 30.

Find the area of the triangle whose two adjacent sides are determined by the vectors

and


Answer:

The area of the triangle = , where a and b are it’s adjacent sides vectors.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 31.

Find the area of the triangle whose two adjacent sides are determined by the vectors

and .
Answer:

The area of the triangle = , where a and b are it’s adjacent sides vectors.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 32.

Using vectors, find the area of ΔABC whose vertices are

A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)
Answer:

Through the vertices we get the adjacent vectors as,



The area of the triangle = , where a and b are it’s adjacent sides vectors.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 33.

Using vectors, find the area of ΔABC whose vertices are

A(1, 2, 3), B(2, −1, 4) and C(4, 5, Δ1) ((considering Δ1 as 1 ))
Answer:

Through the vertices we get the adjacent vectors as,



The area of the triangle = , where a and b are it’s adjacent sides vectors.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 34.

Using vectors, find the area of ΔABC whose vertices are

A(3, −1, 2), B(1, −1, −3) and C(4, −3, 1)
Answer:

Through the vertices we get the adjacent vectors as,



The area of the triangle = , where a and b are it’s adjacent sides vectors.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 35.

Using vectors, find the area of ΔABC whose vertices are

A(1, −1, 2), B(2, 1, −1) and C(3, −1, 2).
Answer:

Through the vertices we get the adjacent vectors as,



The area of the triangle = , where a and b are it’s adjacent sides vectors.


Area =



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒ sq units
Question 36.

Using vector method, show that the given points A, B, C are collinear:

A(3, −5, 1), B(−1, 0, 8) and C(7, −10, −6)
Answer:

Through the vertices we get the adjacent vectors as,



To prove that A, B, C are collinear we need to prove that


.


So,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒



Question 37.

Using vector method, show that the given points A, B, C are collinear:

A(6, −7, −1), B(2, −3, 1) and C(4, −5, 0).
Answer:

Through the vertices we get the adjacent vectors as,



To prove that A, B, C are collinear we need to prove that


.


So,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


Thus, A, B and C are collinear.
Question 38.

Show that the point A, B, C with position vectors , and respectively are collinear.
Answer:

Through the vertices we get the adjacent vectors as,



To prove that A, B, C are collinear we need to prove that


.


So,



Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


Thus, A, B and C are collinear.
Question 39.

Show that the points having position vectors are collinear, whatever be .
Answer:

Through the vertices we get the adjacent vectors as,



To prove that A, B, C are collinear we need to prove that


.


So,


Here,


We


have


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


Thus, A, B and C are collinear.
Question 40.

Show that the points having position vector , and are collinear, whatever be .
Answer:

We have,


Through the vertices we get the adjacent vectors as,



To prove that A, B, C are collinear we need to prove that


.


So,


Here,


We


have



Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


Thus, A, B and C are collinear.
Question 41.

Find a unit vector perpendicular to the plane ABC, where the points A, B, C, are , and respectively.
Answer:

A unit vector perpendicular to the plane ABC will be,



Through the vertices we get the adjacent vectors as,




Here,


We


have


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒


⇒



Question 42.

If and then find .
Answer:

and


Then,


We have,


Here,


We


have
and


⇒


Thus, substituting the values of ,


in equation (i) we get


⇒


⇒



Question 43.

If , and , find .
Answer:

We have,


So,


⇒


⇒



Question 44.

If , and , find the angle between and .
Answer:

We have,


⇒


⇒


⇒


⇒


⇒


⇒