SATHEE: Vectors 5 Question 1

Vectors 5 Question 1

1. Let $α = (λ - 2) a + b$ and $β = (4 λ - 2) a + 3 b$ be two given vectors where vectors $a$ and $b$ are non-collinear. The value of $λ$ for which vectors $α$ and $β$ are collinear, is

(2019 Main, 10 Jan II)

(a) 4

(b) -3

(d) -4

Show Answer

Answer:

Correct Answer: 1. (d)

Solution:

Two vectors $c$ and $d$ are said to be collinear if we can write $c = λ b$ for some non-zero scalar $λ$ .

Let the vectors $α = (λ - 2) a + b$

and

$β = (4 λ - 2) a + 3 b are$

collinear, where $a$ and $b$ are non-collinear.

$∴$ We can write

$\begin{array}{cc} α = k β, for some k \in R - 0 \\ \Rightarrow & (λ - 2) a + b = k [(4 λ - 2) a + 3 b] \\ \Rightarrow & [(λ - 2) - k (4 λ - 2)] a + (1 - 3 k) b = 0 \end{array}$

Now, as $a$ and $b$ are non-collinear, therefore they are linearly independent and hence $(λ - 2) - k (4 λ - 2) = 0$ and $1 - 3 k = 0$

$\Rightarrow λ - 2 = k (4 λ - 2)$ and $3 k = 1$

$\Rightarrow λ - 2 = \frac{1}{3} (4 λ - 2) [∵ 3 k = 1 \Rightarrow k = \frac{1}{3}]$

$\Rightarrow 3 λ - 6 = 4 λ - 2$

$\Rightarrow λ = - 4$

Vectors 5 Question 1

1. Let $α = (λ - 2) a + b$ and $β = (4 λ - 2) a + 3 b$ be two given vectors where vectors $a$ and $b$ are non-collinear. The value of $λ$ for which vectors $α$ and $β$ are collinear, is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Vectors 5 Question 1

1. Let α=(λ−2)a+b and β=(4λ−2)a+3b be two given vectors where vectors a and b are non-collinear. The value of λ for which vectors α and β are collinear, is

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

1. Let $α = (λ - 2) a + b$ and $β = (4 λ - 2) a + 3 b$ be two given vectors where vectors $a$ and $b$ are non-collinear. The value of $λ$ for which vectors $α$ and $β$ are collinear, is