Let a be a vector which is perpendicular to thevector 3-12-2k- If a-2-k-2-13-4k- then the projection of the vector a on the vector 2-2-k- is -

Let a=a_1î+a_2ĵ+a_3k̂ and a. ( 3î 12ĵ+2k̂ )=0⇒ 3a_1+a_22+2a_3=0 nbsp;...(i) and a× (2î+k̂)=2î 13ĵ 4k̂ ⇒ a_2î+(2a_3 a_1)ĵ 2a_2k̂=2î 13ĵ 4k̂ ∴ a_2=2 nbsp; . ...

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Standard XIII

Mathematics

Vector Triple Product

Let a be a ve...

Question

Let $\to a$ be a vector which is perpendicular to thevector $3^i + \frac{1}{2}^j + 2^k .$ If $\to a \times (2^i +^k) = 2^i - 13^j - 4^k,$ then the projection of the vector $a$ on the vector $2^i + 2^j +^k$ is :

1

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\frac{7}{3}

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\frac{1}{3}

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\frac{5}{3}

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Solution

The correct option is D $\frac{5}{3}$
Let $\to a = a_{1}^i + a_{2}^j + a_{3}^k$
and $\to a . (3^i - \frac{1}{2}^j + 2^k) = 0 \Rightarrow 3 a_{1} + \frac{a_{2}}{2} + 2 a_{3} = 0 . . . (i)$

and $\to a \times (2^i +^k) = 2^i - 13^j - 4^k$
$\Rightarrow a_{2}^i + (2 a_{3} - a_{1})^j - 2 a_{2}^k = 2^i - 13^j - 4^k$
$∴ a_{2} = 2$ ...(ii)
and $a_{1} - 2 a_{3} = 13$ ...(iii)
From eq. (i) and (ii) : $a_{1} = 3$ and $a_{3} = - 5$
$^∴ \to a = 3^i + 2^j - 5^k$

$∴$ Projection of $\to a$ on $2^i + 2^j +^k = \frac{6 + 4 - 5}{3} = \frac{5}{3}$

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Let →a be a vector which is perpendicular to thevector 3^i+12^j+2^k. If →a×(2^i+^k)=2^i−13^j−4^k, then the projection of the vector a on the vector 2^i+2^j+^k is :

Let $\to a$ be a vector which is perpendicular to thevector $3^i + \frac{1}{2}^j + 2^k .$ If $\to a \times (2^i +^k) = 2^i - 13^j - 4^k,$ then the projection of the vector $a$ on the vector $2^i + 2^j +^k$ is :