Mr. 'X' wishes to purchase a house for ₹ 49,65,000 with a down payment of ₹ 15,00,000 and balance amount in EMI for 25 years. If bank charges 6% per…

The vector equation of a line passing through two points is $\vec{x} = \vec{a} + \lambda\vec{d}$x=a+λd, where $\vec{a}$a is a position vector of any point on the line, $\vec{d}$d is the direction vector, and $\lambda$λ is a parameter.

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The direction vector is found by subtracting the coordinates of the two points:

$$\vec{d} = B - A$$d=B−A

$$\vec{d} = (1-3)\vec{i} + (-1-4)\vec{j} + (6-(-7))\vec{k}$$d=(1−3)i+(−1−4)j​+(6−(−7))k

$$\vec{d} = -2\vec{i} - 5\vec{j} + 13\vec{k}$$d=−2i−5j​+13k

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Using point $A(3, 4, -7)$A(3,4,−7) as the position vector:

$$\vec{x} = (3\vec{i} + 4\vec{j} - 7\vec{k}) + \lambda(-2\vec{i} - 5\vec{j} + 13\vec{k})$$x=(3i+4j​−7k)+λ(−2i−5j​+13k)

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Expanding the equation:

$$\vec{x} = 3\vec{i} + 4\vec{j} - 7\vec{k} - 2\lambda\vec{i} - 5\lambda\vec{j} + 13\lambda\vec{k}$$x=3i+4j​−7k−2λi−5λj​+13λk

Grouping terms:

$$\vec{x} = (3 - 2\lambda)\vec{i} + (4 - 5\lambda)\vec{j} + (-7 + 13\lambda)\vec{k}$$x=(3−2λ)i+(4−5λ)j​+(−7+13λ)k

Therefore, the vector equation of the line is $\vec{x} = (3 - 2\lambda)\vec{i} + (4 - 5\lambda)\vec{j} + (-7 + 13\lambda)\vec{k}$x=(3−2λ)i+(4−5λ)j​+(−7+13λ)k