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

As d⃗ which is perpendicular to both c⃗ and b⃗ ∴d⃗|| c⃗×b⃗ i.e., d⃗= λ (c⃗×b⃗) ⇒d⃗ = λ[ î amp;ĵ amp;k̂; 3 amp;1 amp; 1; 1 amp; 4 amp; ...

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Byju's Answer

Standard XII

Mathematics

Applications Scalar Triple Product

Let a =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

As $\to d$ which is perpendicular to both $\to c$ and $\to b$

$∴ \to d | | \to c \times \to b$ i.e., $\to d = λ (\to c \times \to b)$

$\Rightarrow \to d = λ ⎡ ⎢ ⎣ \begin{matrix} ^i & ^j & ^k 3 & 1 & - 1 1 & - 4 & 5 \end{matrix} ⎤ ⎥ ⎦ = λ (^i - 16^j - 13^k)$

As $\to d . \to a = 21$ so, $λ (^i - 16^j - 13^k) . (4^i + 5^j -^k) = 21$

$\Rightarrow λ (4 - 80 + 13) = 21 λ = - \frac{1}{3}$

Hence $\to d = - \frac{1}{3} (^i - 16^j - 13^k) = \frac{16^j + 13^k -^i}{3}$

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