Class 12 RD Sharma Solutions – Chapter 25 Vector or Cross Product – Exercise 25.1 | Set 2
Question 13. If , and , find
Solution:
We know that,
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Also,
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And
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Question 14. Find the angle between 2 vectors and , if
Solution:
Given
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=>, as is a unit vector.
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Question 15. If , then show that , where m is any scalar.
Solution:
Given that
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Using distributive property,
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If two vectors are parallel, then their cross-product is 0 vector.
=> and are parallel vectors.
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Hence proved.
Question 16. If, and , find the angle between and
Solution:
Given that,, and
We know that,
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Question 17. What inference can you draw if and
Solution:
Given, and
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Either of the following conditions is true,
1.
2.
3.
4.is parallel to
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Either of the following conditions is true,
1.
2.
3.
4. is perpendicular to
Since both these conditions are true, that implies atleast one of the following conditions is true,
1.
2.
3.
Question 18. If , and are 3 unit vectors such that , and . Show that ,and form an orthogonal right handed triad of unit vectors.
Solution:
Given, , and
As,
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=> is perpendicular to both and .
Similarly,
=> is perpendicular to both and
=> is perpendicular to both and
=> , and are mutually perpendicular.
As, , and are also unit vectors,
=> , and form an orthogonal right-handed triad of unit vectors
Hence proved.
Question 19. Find a unit vector perpendicular to the plane ABC, where the coordinates of A, B, and C are A(3, -1, 2), B(1, -1, 3), and C(4, -3, 1).
Solution:
Given A(3, -1, 2), B(1, -1, 3) and C(4, -3, 1).
Let,
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Plane ABC has two vectors and
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A vector perpendicular to both and is given by,
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To find the unit vector,
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Question 20. If a, b and c are the lengths of sides BC, CA and AB of a triangle ABC, prove that and deduce that
Solution:
Given that , and
From triangle law of vector addition, we have
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Similarly,
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Hence proved.
Question 21. If and , then find . Verify that and are perpendicular to each other.
Solution:
Given, and
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Two vectors are perpendicular if their dot product is zero.
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Hence proved.
Question 22. If and are unit vectors forming an angle of , find the area of the parallelogram having and as its diagonals.
Solution:
Given and forming an angle of .
Area of a parallelogram having diagonals and is
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Thus area is,
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area =
=> Area = square units
Question 23. For any two vectors and , prove that
Solution:
We know that,
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Hence proved.
Question 24. Define and prove that , where is the angle between and
Solution:
Definition of : Let and be 2 non-zero, non-parallel vectors. Then , is defined as a vector with the magnitude of , and which is perpendicular to both the vectors and .
We know that,
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=> ……………..(eq.1)
And as,
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Substituting in (eq.1),
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