Let p-q -r -q-r q - x2 - y2 q - 14 - 4x- 6y p and r-r p - r where p and q are two non-zero non-collinear vectors- and x and y are scalars- Then the value of x-y is

(p× nbsp;q) × nbsp;r +(q·r) nbsp;q = (x^2 + y^2) nbsp;q + (14 4x 6y) nbsp;p ⇒(p·r) nbsp;q (q·r) nbsp;p + (q·r) nbsp;q = (x^2 + y^2 ) nbsp ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Unit Vectors

Let p×q ×r +q...

Question

Let $(\to p \times \to q) \times \to r + (\to q \cdot \to r) \to q = (x^{2} + y^{2}) \to q + (14 - 4 x - 6 y) \to p$ and $(\to r \cdot \to r) \to p = \to r$ where $\to p$ and $\to q$ are two non-zero non-collinear vectors, and $x$ and $y$ are scalars. Then the value of $(x + y)$ is

Open in App

Solution

$(\to p \times \to q) \times \to r + (\to q \cdot \to r) \to q = (x^{2} + y^{2}) \to q + (14 - 4 x - 6 y) \to p$
$\Rightarrow (\to p \cdot \to r) \to q - (\to q \cdot \to r) \to p + (\to q \cdot \to r) \to q = (x^{2} + y^{2}) \to q + (14 - 4 x - 6 y) \to p$
$∴ \to p \cdot \to r + \to q \cdot \to r = x^{2} + y^{2} \dots (1)$
and $- \to q \cdot \to r = 14 - 4 x - 6 y \dots (2)$
From $(1) + (2),$
$\to p \cdot \to r = x^{2} + y^{2} - 4 x - 6 y + 14 \dots (3)$

$(\to r \cdot \to r) \to p = \to r$
Taking dot product with $\to r$ , we get
$(\to r \cdot \to r) (\to p \cdot \to r) = \to r \cdot \to r \Rightarrow \to p . \to r = 1$
$∴$ From equation $(3),$
$x^{2} + y^{2} - 4 x - 6 y + 14 = 1$
$\Rightarrow (x - 2)^{2} + (y - 3)^{2} = 0 \Rightarrow x = 2, y = 3$
Hence, $x + y = 5$

Suggest Corrections

Similar questions

Q. For non-zero vectors

\to p, \to q

and

\to r,

let

(\to p \times \to q) \times \to r + (\to q \cdot \to r) \to q = (x^{2} + y^{2}) \to q + (14 - 4 x - 6 y) \to p

and

(\to r \cdot \to r) \to p = \to r

where

\to p

and

\to q

are non-collinear, and

x

and

y

are scalars. Then the value of

(x + y)

Suppose that p,q and r are three non-coplanar vectors in $R^{3}$ . Let the components of a vector s along p, q and r be 4, 3 and 5 respectively. If the components of this vector s along (-p+q+r), (p-q+r) and (-p-q+r) are $x$ , $y$ and $z$ respectively, then the value of $2 x + y + z$ is ___

Suppose that p,q and r are three non-coplanar vectors in $R^{3}$ . Let the components of a vector s along p, q and r be 4, 3 and 5 respectively. If the components of this vector s along (-p+q+r), (p-q+r) and (-p-q+r) are $x$ , $y$ and $z$ respectively, then the value of $2 x + y + z$ is ___