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The vector vec b=3 hat i+4 hat k is to ...

The vector ` vec b=3 hat i+4 hat k` is to be written as the sum of a vector ` vecalpha` parallel to ` vec a= hat i+ hat j` and a vector ` vecbeta` perpendicular to ` vec adot` Then ` vecalpha=` `3/2( hat i+ hat j)` b. `2/3( hat i+ hat j)` c. `1/2( hat i+ hat j)` d. `1/3( hat i+ hat j)`

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To solve the problem, we need to express the vector \( \vec{b} = 3 \hat{i} + 4 \hat{k} \) as the sum of two vectors: \( \vec{\alpha} \) which is parallel to \( \vec{a} = \hat{i} + \hat{j} \) and \( \vec{\beta} \) which is perpendicular to \( \vec{a} \). ### Step-by-step Solution: 1. **Define the vectors**: - Given \( \vec{b} = 3 \hat{i} + 4 \hat{k} \) - Given \( \vec{a} = \hat{i} + \hat{j} \) ...
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  • The unit vector perpendicular to vec A = 2 hat i + 3 hat j + hat k and vec B = hat i - hat j + hat k is

    A
    `(4hati-hatj-5hatk)/(sqrt(42))`
    B
    `(4hati-hatj+5hatk)/(sqrt(42))`
    C
    `(4hati+hatj+5hatk)/(sqrt(42))`
    D
    `(4hati+hatj-5hatk)/(sqrt(42))`

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