If the points A, B, C with position vectors 20i +, 5i - j and 10i - 13j respectively are collinear, then the value of is Option 1: 12 Option 2: -37 Option 3: 37 Option 4: -12

Answer: -37. Solution: If three points are collinear, they lie on the same straight line. This means the vectors connecting them must be parallel, i.e., their cross product equals zero. A = 20i +, B = 5i - j, C = 10i - 13j AB = B - A = (5 - 20)i + (-1 - )j = -15i + (-1 - )j AC = C - A = (10 - 20)i + (-13 - )j = -10i + (-13 - )j For collinearity, the cross product of AB and AC must be zero: a_1 * b_2 - a_2 * b_1 = 0 (-15)(-13 - ) - (-1 - )(-10) = 0 Expanding: (195 + 15) - (10 + 10) = 0 185 + 5 = 0 5 = -185 = -37 Verification with = -37: AB = -15i + 36j AC = -10i + 24j (-15)/(-10) = 3/2 36/24 = 3/2 The ratios are equal, so the points are collinear. ✅ = -37

Algebra

Vector Algebra

Medium

core

If the points $A, B, C$ with position vectors $20\hat{i} + \lambda\hat{j}$ , $5\hat{i} - \hat{j}$ and $10\hat{i} - 13\hat{j}$ respectively are collinear, then the value of $\lambda$ is

$12$

$-37$

$37$

$-12$

✅ Correct Option: 2

If three points are collinear, they lie on the same straight line. This means the vectors connecting them must be parallel, i.e., their cross product equals zero.

$A = 20\hat{i} + \lambda\hat{j}$ , $B = 5\hat{i} - \hat{j}$ , $C = 10\hat{i} - 13\hat{j}$

$\vec{AB} = B - A = (5 - 20)\hat{i} + (-1 - \lambda)\hat{j} = -15\hat{i} + (-1 - \lambda)\hat{j}$

$\vec{AC} = C - A = (10 - 20)\hat{i} + (-13 - \lambda)\hat{j} = -10\hat{i} + (-13 - \lambda)\hat{j}$

For collinearity, the cross product of $\vec{AB}$ and $\vec{AC}$ must be zero:

$a_1 \cdot b_2 - a_2 \cdot b_1 = 0$

$(-15)(-13 - \lambda) - (-1 - \lambda)(-10) = 0$

Expanding:

$(195 + 15\lambda) - (10 + 10\lambda) = 0$

$185 + 5\lambda = 0$

$5\lambda = -185$

$\lambda = -37$

Verification with $\lambda = -37$ :

$\vec{AB} = -15\hat{i} + 36\hat{j}$

$\vec{AC} = -10\hat{i} + 24\hat{j}$

$\dfrac{-15}{-10} = \dfrac{3}{2}$

$\dfrac{36}{24} = \dfrac{3}{2}$

The ratios are equal, so the points are collinear. ✅

$\boxed{\lambda = -37}$

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If the points $A, B, C$ with position vectors $20\hat{i} + \lambda\hat{j}$ , $5\hat{i} - \hat{j}$ and $10\hat{i} - 13\hat{j}$ respectively are collinear, then the value of $\lambda$ is