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; ...

You visited us 3 times! Enjoying our articles? Unlock Full Access!

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

Open in App

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

Suggest Corrections

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. 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 \cdot \to a = 21

Q. If

\to a =^i + 2^j +^k, \to b = 2^i +^j

and

\to c = 3^i - 4^j - 5^k

, then find a unit vector perpendicular to both of the vectors

(\to a - \to b)

and

(\to c - \to b)

Q. A unit vector parallel to the resultant of the vectors

\to A =^i + 4^j - 2^k

and

\to B = 3^i - 5^j +^k

Show that the four points A, B, C and D with position vectors $4^i + 5^j +^k, -^j -^k, 3^i + 9^j + 4^k$ and $4 (-^i +^j +^k)$ respectively are coplannar.

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 $\to b + \to c$ .

Join BYJU'S Learning Program

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