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Vector Algebra

Question
CBSEENMA12035731
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If a unit vector straight a with rightwards arrow on top makes angle straight pi over 3 space with space straight i with hat on top comma space space space straight pi over 4 space with space straight j with hat on top space and space acute space angle space straight theta space with space straight k with hat on top comma space then find the value of straight theta.

Solution
Let space straight a with rightwards arrow on top space equals space straight x straight i with hat on top plus straight y straight j with hat on top plus straight z straight k with hat on top space be space the space unit space vector. therefore space space straight x space equals space cos straight pi over 3 equals space 1 half straight y space equals cos straight pi over 4 equals fraction numerator 1 over denominator square root of 2 end fraction Now comma space straight x squared plus straight y squared plus straight z squared space equals space 1 rightwards double arrow space open parentheses 1 half close parentheses squared plus open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses squared plus straight z squared space equals space 1 rightwards double arrow straight z squared space equals space 1 minus 1 fourth minus 1 half equals 1 fourth rightwards double arrow space straight z space equals space 1 half therefore space space cos space straight theta space equals space 1 half space rightwards double arrow space space straight theta space equals space straight pi over 3

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