The Unit Vectors orthogonal to -2-2k- and making equal angles with x and y axes are

The correct option is B ±13(2î+2ĵ k̂) Let orthogonal vector be aî+bĵ+ck̂ (ai+bĵ+ck̂).( î+αĵ+αk̂)=0 —— (1) (both vetors are ⊥ to each othe ...

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Standard XII

Mathematics

Condition for Coplanarity of Four Points

The Unit Vect...

Question

The Unit Vectors orthogonal to $-^i + 2^j + 2^k$ and making equal angles with $x$ and $y$ axes are

\pm \frac{1}{3} (2^i + 2^j -^k)

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\pm \frac{1}{3} (^i +^j -^k)

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\pm \frac{1}{3} (2^i - 2^j -^k)

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\pm \frac{1}{4} (^i +^j -^k)

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Solution

The correct option is B $\pm \frac{1}{3} (2^i + 2^j -^k)$
Let orthogonal vector be $a^i + b^j + c^k$
$(a i + b^j + c^k) . (-^i + α^j + α^k) = 0$ -------(1) (both vetors are $⊥$ to each other)
As they make equal angle with $x$ & $y$
So $a = b$ -----------(2)
& so unit vector $= \frac{a^i + b^j + c^k}{\sqrt{a^{2} + b^{2} + c^{2}}}$ ------------(3)
So all condition is satisfied in option $A .$

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The Unit Vectors orthogonal to −^i+2^j+2^k and making equal angles with x and y axes are

The Unit Vectors orthogonal to $-^i + 2^j + 2^k$ and making equal angles with $x$ and $y$ axes are