Class 12 RD Sharma Solutions – Chapter 25 Vector or Cross Product – Exercise 25.1 | Set 1
Question 1. If and , find
Solution:
Given, and .
=> =
=> =
=> =
=> =
=> =
Now,
=> =
=> =
=> = √91
Question 2(i). If and , find the value of
Solution:
Given, and
=> =
=> =
=> =
=> =
=> =
Now,
=> =
=> =
=> =
Question 2(ii). If and , find the magnitude of
Solution:
Given, and
=> =
=> =
=> =
=> =
=> =
Now,
=> =
=> =
=> = √6
Question 3(i). Find a unit vector perpendicular to both the vectors and
Solution:
Given and
A vector perpendicular to 2 vectors is given by
=> =
=> =
=> =
=> =
=> =
Unit vector is given by
=> =
=> =
=> = 3
=> Unit vector is,
=> =
Question 3(ii). Find a unit vector perpendicular to the plane containing the vectors and .
Solution:
Given, and
A vector perpendicular to 2 vectors is given by
=> =
=> =
=> =
=> =
=> =
Unit vector is given by
=> =
=> =
=> =
=> Unit vector is,
=> =
Question 4. Find the magnitude of vector
Solution:
Given
=>
=> =
=> =
=> =
=> =
Unit vector is,
=> =
=> =
=> = √74
Question 5. If and , then find
Solution:
Given, and
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
Now,
=> =
=> =
=> =
Question 6. If and , find
Solution:
Given, and
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
Question 7(i). Find a vector of magnitude 49, which is perpendicular to both the vectors and
Solution:
Given, and
A vector perpendicular to 2 vectors is given by
=> =
=> =
=> =
=> =
=> =
Magnitude of vector is given by,
=> =
=> =
=> =
=> =
=> Vector is,
Question 7(ii). Find the vector whose length is 3 and which is perpendicular to the vector and
Solution:
Given, and
A vector perpendicular to 2 vectors is given by
=> =
=> =
=> =
=> =
=> =
Magnitude of vector is given by,
=> =
=> =
=> =
=> = 27
=> Unit vector is,
=> =
=> =
Required vector is,
=>
Question 8(i). Find the parallelogram determined by the vectors: and
Solution:
Given that, and
=> Area of the parallelogram is
=> =
=> =
=> =
=> =
=> =
Thus the area of parallelogram is,
=> =
=> =
=> Area = 6 square units.
Question 8(ii). Find the parallelogram determined by the vectors: and .
Solution:
Given that, and
=> Area of the parallelogram is
=> =
=> =
=> =
=> =
=> =
Thus, the area of parallelogram is,
=> =
=> =
=> Area =
Question 8(iii). Find the area of the parallelogram determined by the vectors: and
Solution:
Given that, and
=> Area of the parallelogram is
=> =
=> =
=> =
=> =
=> =
Thus the area of parallelogram is,
=> =
=> =
=> Area =
Question 8(iv). Find the area of the parallelogram determined by the vectors: and
Solution:
Given that, and
=> Area of the parallelogram is
=> =
=> =
=> =
=> =
=> =
Thus the area of parallelogram is,
=> =
=> =
=> Area =
Question 9(i). Find the area of the parallelogram whose diagonals are: and
Solution:
Given, and
=> Area of the parallelogram is
=> =
=> =
=> =
=> =
=> =
Thus the area of parallelogram is,
=> =
=> =
=> Area = 15/2 = 7.5 square units
Question 9(ii). Find the area of the parallelogram whose diagonals are: and
Solution:
Given, and
=> Area of the parallelogram is
=> =
=> =
=> =
=> =
=> =
Thus the area of parallelogram is,
=> =
=> =
=> Area =
Question 9(iii). Find the area of the parallelogram whose diagonals are: and
Solution:
Given, and
=> Area of the parallelogram is
=> =
=> =
=> =
=> =
=> =
Thus the area of parallelogram is,
=> =
=> =
=> Area =
Question 9(iv). Find the area of the parallelogram whose diagonals are: and
Solution:
Given, and
=> Area of the parallelogram is
=> =
=> =
=> =
=> =
=> =
Thus the area of parallelogram is,
=> =
=> =
=> Area =
=> Area = 24.5
Question 10. If , and , compute and and verify these are not equal.
Solution:
Given , and
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> =
=> is not equal to
=> Hence verified.
Question 11. If , and , find
Solution:
We know that,
=>
=>
We know that is 1, as is a unit vector
=>
=>
=>
Also,
=>
And
=>
=>
=>
=>
=>
=>
Question 12. Given , , , , , being a right-handed orthogonal system of unit vectors in space, show that , and is also another system.
Solution:
To show that , and is a right-handed orthogonal system of unit vectors, we need to prove:
(1)
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
=>
(2)
=>
=>
=>
=>
=>
(3)
=>
=>
=>
=>
=>
(4)
=>
=>
=>
=>
=>
Hence proved.
Last Updated :
28 Mar, 2021
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