Vector 2-4- 3-6- and -2-5- have same initial point 2-2- then the value of 2- for which all three vectors vanishes on a straight line is

Let the initial point is P(2, 2) and vectors PA=2î+4ĵ, PB=3î+6ĵ, PC=(λ 2)î+5ĵ then for terminal point, PA=OA OP ⇒OA=PA+OP=4î+2ĵ similarly, OB=PB+OP=5î+4ĵ and OC ...

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Byju's Answer

Standard IX

Mathematics

Equation of a Plane Passing through a Point and Perpendicular to a Given Vector

Vector 2î+4ĵ,...

Question

Vector $2^i + 4^j, 3^i + 6^j$ and $(λ - 2)^i + 5^j$ have same initial point $(2, - 2),$ then the value of $2 λ$ for which all three vectors vanishes on a straight line is

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Solution

Let the initial point is $P (2, - 2)$ and vectors $- - \to P A = 2^i + 4^j, - - \to P B = 3^i + 6^j, - - \to P C = (λ - 2)^i + 5^j$
then for terminal point, $- - \to P A = - - \to O A - - - \to O P$
$\Rightarrow - - \to O A = - - \to P A + - - \to O P = 4^i + 2^j$
similarly, $- - \to O B = - - \to P B + - - \to O P = 5^i + 4^j$ and
$- - \to O C = - - \to P C + - - \to O P = λ^i + 3^j$
for collinear points of $A, B, C$ slope of $A B =$ slope of $A C$
$\Rightarrow \frac{4 - 2}{5 - 4} = \frac{3 - 2}{λ - 4} \Rightarrow λ - 4 = \frac{1}{2} ∴ 2 λ = 9$

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Vector 2^i+4^j,3^i+6^j and (λ−2)^i+5^j have same initial point (2,−2), then the value of 2λ for which all three vectors vanishes on a straight line is

Vector $2^i + 4^j, 3^i + 6^j$ and $(λ - 2)^i + 5^j$ have same initial point $(2, - 2),$ then the value of $2 λ$ for which all three vectors vanishes on a straight line is