Determine the vector equation for the line, given the Cartesian equation is x+5/3 = y-7/2 = z+3/2 ? - Mathematics MCQ Mathematics Mock Test - 8 2026 | ExamTest.live

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Determine the vector equation for the line, given the Cartesian equation is \frac{x + 5}{3} = \frac{y - 7}{2} = \frac{z + 3}{2} ?

A $r = (- 3 \hat{i} + 5 \hat{j} - 6 \hat{k}) + λ (2 \hat{i} + 4 \hat{j} + 2 \hat{k})$
B $r = (- 5 \hat{i} + 7 \hat{j} - 3 \hat{k}) + λ (2 \hat{i} + 4 \hat{j} + 2 \hat{k})$
C $r = (- 5 \hat{i} + 7 \hat{j} - 3 \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 2 \hat{k})$
D $r = (- 3 \hat{i} + 5 \hat{j} - 6 \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 2 \hat{k})$

Solution & Step-by-step Explanation

The standard Cartesian form of a line is: \frac{x - x _{1}}{a} = \frac{y - y _{1}}{b} = \frac{z - z _{1}}{c} The vector form is r = a + λ b, where a = x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k} is a point on the line and b = a \hat{i} + b \hat{j} + c \hat{k} is the direction vector.From the given equation: x_{1} = - 5, y_{1} = 7, z_{1} = - 3 a = 3, b = 2, c = 2 Thus: a = - 5 \hat{i} + 7 \hat{j} - 3 \hat{k} b = 3 \hat{i} + 2 \hat{j} + 2 \hat{k} Vector equation: r = (- 5 \hat{i} + 7 \hat{j} - 3 \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 2 \hat{k})

Try it yourself before checking the explanation above.

Determine the vector equation for the line, given the Cartesian equation is \frac{x + 5}{3} = \frac{y - 7}{2} = \frac{z + 3}{2} ?

r = (- 3 \hat{i} + 5 \hat{j} - 6 \hat{k}) + λ (2 \hat{i} + 4 \hat{j} + 2 \hat{k})

r = (- 5 \hat{i} + 7 \hat{j} - 3 \hat{k}) + λ (2 \hat{i} + 4 \hat{j} + 2 \hat{k})

r = (- 5 \hat{i} + 7 \hat{j} - 3 \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 2 \hat{k})

r = (- 3 \hat{i} + 5 \hat{j} - 6 \hat{k}) + λ (3 \hat{i} + 2 \hat{j} + 2 \hat{k})

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