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Distance between Two Parallel Planes

Let a⃗=4î+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 $\to a$ and $\to b$ , and is such that $\to d \cdot \to c = 21$ .

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Solution

Given to us that,

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

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

$\to c = 3 i + j - k$

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

Therefore according to question:-

$\to d . \to c = 21$

$(x i + y j + z k) . (3 i + j - k) = 21 (b e c a u s e i . i = j . j = k . k = 1)$

On solving the above equation we get $3 x + y - z = 21$ ................(let equation 1)
We know that when two vectors are perpendicular then dot product of two vectors is equal to zero

Therefore $\to d . \to a = 0$

$(x i + y j + z k) . (4 i + 5 j - k) = 0$

i.e. $4 x + 5 y - z = 0$ ...................(let equation 2)

Therefore $\to d . \to b = 0$

$(x i + y j + z k) . (i - 4 j + 5 k) = 0$

i.e. $x - 4 y + 5 z = 0$ ..................(let equation 3)

Now on adding equation1 and equation 3 and subtract it sum from equation 2
We get $8 y - 5 z = - 21$ .................(let equation 4)

On multiplying equation 3 by 3 and subtract it from equation 1
We get $13 y - 16 z = 21$ ................(let equation 5)

On solving equation 4 and equation 5

We get $y = z = - 7$

Now on putting value of y and z in equation equation 3

We get $x = 7$

Therefore $\to d = 7 i - 7 j - 7 k$

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Similar questions

\to a =^i + 4^j + 2^k, \to b = 3^i - 2^j + 7^k

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

. Find a vector

\to d

which is perpendicular to both

\to a

\to b

\to c \cdot \to d = 15

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

\to c

be a unit vector perpendicular to

\to a

and coplanar with

\to a

\to b

\to c

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

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

, find the unit vector along any vector parallel to the resultant of

\to A

\to B

Q. Find the cosine of the angle between the directions of the vectors

\to a = 4^i + 3^j +^k, \to b = 2^i -^j + 2^k .

Also find a unit vector perpendicular to both

\to a

\to b

. What is the sine of the angle between

\to a

\to b

\to a

\to b

are vectors such that

| \to a + \to b | = \sqrt{29}

\to a \times (2^i + 3^j + 4^k) = (2^i + 3^j + 4^k) \times \to b

, then a possible value of

(\to a + \to b) . (- 7^i + 2^j + 3^k)

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