Let a-4 -5 -k- b-4 -5 k- and c-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

Mathematics

Definition of Vector

Let a⃗=4 î+5 ...

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

Q. Find the vector

\to p

which is perpendicular to both

\to a = 4^i + 5^j -^k

and

\to b =^i - 4^j + 5^k

and

\to p \cdot \to q = 21

and

\to q = 3^i +^j -^k

Q. Vectors

\to a = 3^i +^j +^k, \to b =^i -^j + 2^k a n d \to c = 2^i -^j -^k

. Find vector

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and

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Q. If

\to a =^i +^j +^k

\to b =^i -^j +^k

\to c =^i +^j -^k

and

\to a =^i -^j -^k

, then

(\to a \times \to b) \times (\to c \times \to d)

is a vector orthogonal to both

Q. If

\to a =^i +^j +^k, \to b =^i +^j -^k

\to c =^i -^j +^k

and

\to d =^i -^j -^k

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(\to a \times \to b) \times (\to c \times \to d) =

Q. If

\to a = 2^i +^j + 3^k

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, then

(\to a \times \to b) \times (\to c \times \to d) =

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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 $\to d$ . $\to a = 21$ .