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Addition of Vectors

Let a̅,b̅ b...

Question

Let $¯ a, ¯ b$ be two noncollinear vectors. If $A = (x + 4 y) ¯ a + (2 x + y + 1) ¯ b,$
$B = (y - 2 x + 2) ¯ a + (2 x - 3 y - 1) ¯ b a n d 3 A = 2 B$ then (x,y) =

A

(1,2)

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B

(1,-2)

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C

(2,-1)

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D

(-2,-1)

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Solution

The correct option is A (2,-1)
$3 A = (3 x + 12 y) ¯ a + (6 x + 3 y + 3) ¯ b$
$2 B = (2 y - 4 x + 4) ¯ a + (4 x - 6 y - 2) ¯ b$
$3 A = 2 B$
$\Rightarrow 3 x + 12 y = 2 y - 4 x + 4$ & $6 x + 3 y + 3 = 4 x - 6 y - 2$
$7 x + 10 y = 4$ & $2 x + 9 y = - 5$
Solving $7 x + 10 y = 4$ & $2 x + 9 y = - 5$
We get $x = 2$ , $y = - 1$
$∴ (x, y) = (2, - 1)$ .

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Similar questions

a, b

be two noncollinear vectors. If

A = (x + 4 y) a + (2 x + y + 1) b

B = (y - 2 x + 2) a + (2 x - 3 y - 13) b

3 A = 2 B

(x, y)

=

Q. Let a,b be two noncoffinear vectors. If

¯ ¯¯¯¯¯¯ ¯ O A = (x + 4 y) ¯ ¯ ¯ a + (2 x + y + 1) ¯ ¯ b, ¯ ¯¯¯¯¯¯ ¯ O B = (y - 2 x + 2) ¯ ¯ ¯ a + (2 x - 3 y - 1) ¯ ¯ b

3 ¯ ¯¯¯¯¯¯ ¯ O A = 2 ¯ ¯¯¯¯¯¯ ¯ O B,

(x, y) =

¯ a

¯ b

are non collinear vectors. If

¯ c = (x - 2) ¯ a + ¯ b

¯ d = (2 x + 1) ¯ a - ¯ b

are collinear vectors, then find the value of

x

¯ ¯¯ ¯ α = (x + 4 y) a + (2 x + y + 1) b

¯ ¯ ¯ β = y - 2 x + 2) a + (2 x - 3 y - 1) b

b

are nonzero, noncollinear vectors if 3

¯ ¯¯ ¯ α = 2 ¯ ¯ ¯ β = x = 2, y = - 1

D, E, F

be the middle points of the sides

B C . C A, A B

respectively of

Δ A B C

A D + B E + C F = 0

.

I .

3 x - 5 y = - 1

x - y = - 1

x = - 2, y = - 1

I I .

2 x + 3 y = 9, 3 x + 4 y = 5 \Rightarrow x = - 21, y = 17

I I I .

\frac{2 x}{a} + \frac{y}{b} = 2, \frac{x}{a} - \frac{y}{b} = 4 \Rightarrow x = 2 a, y = 2 b

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