a) find the value of a when vector 3i+2j +9k and i+ aj +3K are
1)Perpendicular 2) Parallel
Answers
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Answer:
When vectors are (i) Perpendicular, the value of a = -15
(ii) Parallel, the value of a = 2/3
Step-by-step explanation:
Given two vectors are
3i+2j+9k and i +aj+3k
- Let A = 3i+2j+9k and B = i +aj+3k
- If the two vectors are Perpendicular then their dot product is zero.
A.B = 0
(3i+2j+9k).(i +aj+3k) = 0
(3i).(i)+(2j).(aj)+(9k).(3k) = 0
3(i.i)+2a(j.j)+27(k.k) = 0
3+2a+27 = 0
2a+30 = 0
2a = -30
a = -15
When the given vectors are Perpendicular then a = -15.
- If the two vectors are Parallel then their cross product is equal to zero.
AXB = 0
(3i+2j+9k) X (i +aj+3k) = 0
We know that
AXB = (a₂b₃-b₂a₃)i - (a₁b₃-a₃b₁)j + (a₁b₂-a₂b₁)k
where a₁ = 3, a₂ = 2, a₃ = 9,
b₁ = 1, b₂ = a, b₃ = 3
AXB = [(2×3)-(a×9)]i - [(3×3)-(9×1)]j+[(3×a)-(2×1)]k = 0
(6-9a)i - (9-9)j + (3a-2)k = 0
Now take 6-9a = 0
9a = 6
a = 6/9
a = 2/3
When the given vectors are Parallel then a = 2/3.
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