A unit vector in the x y -plane that makes an angle of 45 ° with the vector i ˙ ^ + j ˙ ^ and an angle of 60 ° with the vector 3 i ˙ ^ - 4 j ˙ ^ is

Question

A unit vector in the x y -plane that makes an angle of 45 ° with the vector i ˙ ^ + j ˙ ^ and an angle of 60 ° with the vector 3 i ˙ ^ - 4 j ˙ ^ is, i ˙ ^ + j ˙ ^ 2, i ˙ ^ - j ˙ ^ 2, 2 i ˙ ^ - j ˙ ^ 2, None of these

MARKS App · Accepted Answer

Let the required unit vector be r → = a i ˙ ^ + b j ˙ ^ Then, r → = 1 ⟹ a 2 + b 2 = 1 ...(i) Since, r → makes an angle of 45 ° with i ˙ ^ + j ˙ ^ and an angle of 60 ° with 3 i ˙ ^ - 4 j ˙ ^ , therefore cos ⁡ π 4 = r → ⋅ ( i ˙ ^ + j ˙ ^ ) r → | i ˙ ^ + j ˙ ^ | and cos ⁡ π 3 = r → ⋅ 3 i ˙ ^ - 4 j ˙ ^ r → 3 i ˙ ^ - 4 j ˙ ^ ⟹ 1 2 = a + b 2 a n d 1 2 = 3 a - 4 b 5 ⟹ a + b = 1 a n d 3 a - 4 b = 5 2 ⟹ a = 13 14 , b = 1 14 ∴ r → = 13 14 i ˙ ^ + 1 14 j ˙ ^

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