Question

if r=3i+2j-5k a=2i-j+k b=i+3j-2k c=-2i+j-3k such that r=la+mb+nc then
ple give write ans
here the options are a)m,1/2,n are in ap
b)l,m,n are in ap
c)l,m,n are in hp
d)m,l,n are in gp

Answer 1

Step-by-step explanation:

Option (A) must be m, l/2, n are in A. P

Answer 2

The correct answer is (a) m, l/2, n are in AP.

Given: r = 3i + 2j - 5k, a = 2i - j + k, b = i + 3j - 2k, c = -2i + j - 3k

r = la + mb + nc

To Find: The relation between l, m, n.

Solution:

We are given the values of r, a, b, c as,

r = 3i + 2j - 5k, a = 2i - j + k, b = i + 3j - 2k, c = -2i + j - 3k

We also have, r = la + mb + nc                         ...... (1)

Putting respective values in (1),  we get;

          r = la + mb + nc    

  ⇒ 3i + 2j - 5k = l ( 2i - j + k ) + b ( i + 3j - 2k ) + c ( -2i + j - 3k )

The vector form can also be written in form of point,

⇒ ( 3, 2, - 5 ) = l ( 2, - 1, 1 ) + m ( 1, 3, - 2 ) + n ( -2, 1, - 3 )

Comparing the values we get 3 equations in term of l, m, n.

2l + m - 2n = 3                                    ...... (2)

- l + 3m + n = 2                                     ...... (3)

l - 2m - 3n = - 5                                    ....... (4)

According to the 3 equations, they can be written in matrix form like,

$\left[\begin{array}{ccc}2&1&-2\\-1&3&1\\1&-2&-3\end{array}\right]$ $\left[\begin{array}{ccc}l\\m\\n\end{array}\right] = \left[\begin{array}{ccc}3\\2&\\-5\end{array}\right]$

R2 → R2 + R3

$\left[\begin{array}{ccc}2&1&-2\\0&1&-2\\1&-2&-3\end{array}\right] \left[\begin{array}{ccc}l\\m\\n\end{array}\right] = \left[\begin{array}{ccc}3\\-3\\-5\end{array}\right]$

R1 → R1 - R2

$\left[\begin{array}{ccc}2&0&0\\0&1&-2\\1&-2&-3\end{array}\right] \left[\begin{array}{ccc}l\\m\\n\end{array}\right] = \left[\begin{array}{ccc}6\\-3\\-5\end{array}\right]$

Multiplying the matrices we get,

   2×l = 6

⇒ l = 3                                   ...... (5)

   m - 2n = - 3                        ..... (6)

   l - 2m - 3n = -5                   ..... (7)

Solving (5), (6), (7), we get,

l = 3, m = 1, n = 2

Now for getting the desired relation between l,m, and n we use hit and trial,

(a) m, l/2, n are in AP.

   For the above condition to be true,

⇒ 2 × l/2 = m + n,  must be true

RHS: 2 × 3/2 = 3

LHS: m + n = 1 + 2 = 3

Thus, RHS = LHS

Hence, the correct answer is (a) m, l/2, n are in AP.

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