Let a-4- 5-k- b- - 4- - 5 k- andc-3- - - - k- Find a vector d which is perpendicular to both c and b and d- a- 21-

Let d= xî+ y ĵ + z k̂ Given that d is perpendicular to both c and b ∴ d·c= 0 ⇒ 3x y z = 0 ....... (i) d·b = 0 ⇒ x 4y + 5z = 0 ....... (ii) nbsp; And ...

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Byju's Answer

Standard XII

Physics

Vectors and Its Types

Let a=4î+ 5ĵ-...

Question

Let $\to a = 4^i + 5^j -^k, \to b =^i - 4^j + 5^k, and \to c = 3^i -^j -^k$ . Find a vector $\to d$ which is perpendicular to both $\to c$ and $\to b$ and $\to d$ . $\to a = 21$ .

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Solution

Let $\to d = x^i + y^j + z^k$

Given that $\to d$ is perpendicular to both $\to c$ and $\to b$

$∴ \to d \cdot \to c = 0 \Rightarrow 3 x - y - z = 0$ ....... $(i)$

$\to d \cdot \to b = 0 \Rightarrow x - 4 y + 5 z = 0$ ....... $(i i)$

And $\to d \cdot \to a = 21 \Rightarrow 4 x + 5 y - z = 21$ ......... $(i i i)$

From $(i i i)$ and $(i)$ , we have

$x + 6 y = 21$ ....... $(i v)$

From $(i)$ and $(i i)$ , we have

$16 x - 9 y = 0 \Rightarrow y = \frac{16}{9} x$ ...... $(v)$

From $(i v)$ and $(v)$ , we obtain

$x = \frac{9}{5}$ and $y = \frac{16}{5}$

From $(i)$ , we have $z = \frac{11}{5}$

Thus, the required vector

$\to d = \frac{9}{5}^i + \frac{16}{5}^j + \frac{11}{5}^k$
or
$= \frac{1}{5} (9^i + 16^j + 11^k)$

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Similar questions

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The scalar product of the vector $\to a =^i +^j +^k$ with a unit vector along the sum of vector $\to b = 2^i + 4^j + 5^k$ and $\to c = λ^i + 2^j + 3^k$ is equal to one. Find the value of $λ$ and hence find the unit vector along $\to b + \to c$ .

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Let →a=4^i+5^j−^k, →b=^i−4^j+5^k, and→c=3^i−^j−^k. Find a vector →d which is perpendicular to both →c and →b and →d. →a=21.

Let $\to a = 4^i + 5^j -^k, \to b =^i - 4^j + 5^k, and \to c = 3^i -^j -^k$ . Find a vector $\to d$ which is perpendicular to both $\to c$ and $\to b$ and $\to d$ . $\to a = 21$ .