A unit vector in the xy plane which makes an angle of 45 with the vector (i^+j^) and an angle of 60 with the vector (3i^4j^) is ____.
Answers
unit vector is (13i + i)/√170
let a unit vector A = x i + y j
so, magnitude of A = √(x² + y²) = 1 ⇒ x² + y² = 1......(1)
a/c to question,
cos45° = (x i + y j).(i + j)/|A||(i + j)|
⇒1/√2 = (x + y)/√(x² + y²) × √2
⇒x + y = √(x² + y²) ...........(2)
and cos60° = (x i + y j).(3i + 4j)/|A||3i - 4j|
⇒1/2 = (3x - 4y)/√(x² + y²) × 5
⇒(3x - 4y) =5/2√(x² + y²)
⇒2/5(3x - 4y) =√(x² + y²) .......(3)
from equations (2) and (3) we get,
(x + y) = 2/5(3x - 4y)
⇒5x + 5y = 6x - 8y
⇒13y =x
now from equation (1),
169y² + y² = 1 ⇒y = ±1/√170
so, x = ± 13/√170
hence, A = ±(13i + y)/√170
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