Important Questions for Class 12 Chapter 10 - Vector algebra

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Important questions based on NCERT syllabus for Class 12 Chapter 10 - Vector algebra:

Question 1: Find a vector in the direction of vector 5i + 4j + 2k which has a magnitude 27 units

Solution: Let a = 5i + 4j + 2k

|a| = √(5)^2 + (4)^2 + (2)^2
|a| = √25 + 16 + 4
|a| = √45
|a| = √(9 * 5)
|a| = 3√5 units

The unit vector in the direction of the given vector a is

$\hat{a} = $\vec{a}/|\vect{a}| = 1/3√5(5\hat{i} + 4j + 2k) = 5/3√5 i + 4/3√5 j + 2/3√5 k

Therefore, the vector of magnitude equal to 32 units and in the direction of a is

32\hap{a} = 27(5/3√5 i + 4/3√5 j + 2/3√5 k)
= 9/√5 i +36/√5 j + 18/√5 k

Question 2: Write the unit vector in the direction of the sum of the vector a = 3i + 2j + 4k and b = i + j + 8k

Solution: Given a = 3i + 2j + 4k and b = i + j + 8k

We have to find a + b

a + b = 3i + 2j + 4k + i + j + 8k
a + b = 4i + 3j + 12k

and |a+b| = √((4)^2 + (3)^2 + (12)^2)
|a+b| = √(16 + 9 + 144)
|a+b| = √(25 + 144)
|a+b| = √169
|a+b| = 13 units.

Therefore, required unit vector = Unit vector along the direction of a+b
= \vec{a}+b/|a+b| = 4/13 i + 3/13 j + 12/13k

Question 3: A and B are two points with position vectors 3a - 2b and 2a + 3b respectively. Write the position vector of a point C which divide line segment AB in the ratio 2:1 externally.

Solution: Given A and B are two points with position vectors 3a - 2b and 2a + 3b respectively.

Also point C which divide line segment AB in the ratio 2:1 externally.

Therefore, position vector of a point C

= [2(3a - 2b) - 1(2a + 3b)]/2-1
= 6a - 4b - 2a - 3b/1
= 4a - 7b