Question
Download Solution PDFDetermine the vector equation for the line, given the cartesian equation of a line is \(\frac{{x + 5}}{3} = \frac{{y - 7}}{2} = \frac{{z + 3}}{2}\)?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Equation of a line in cartesian form:- \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\)
Equation of a line in vector form:- \(\vec r = \vec a + \lambda \vec b\)
where, \(\vec a = x_1 \hat{i} + y_1\hat j + z_1 \hat k\) and \(\vec b = a \hat i + b \hat j + c \hat k\)
Calculation:
We have,
Cartesian equation:- \(\frac{{x + 5}}{3} = \frac{{y - 7}}{2} = \frac{{z + 3}}{2}\)
⇒ \(\frac{{x - (-5)}}{3} = \frac{{y - 7}}{2} = \frac{{z - (-3)}}{2}\) ------- equation (1)
⇒ \(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\) -------- equation (2)
Comparing (1) and (2)
⇒ x1 = -5, y1 = 7, z1 = -3
⇒ a = 3, b = 2, c = 2
⇒ \(\vec a = -5\hat i + 7\hat j - 3 \hat k\)
⇒ \(\vec b = 3 \hat i + 2 \hat j + 2 \hat k\)
Equation of line in vector form:- \(\vec r = \vec a + \lambda \vec b\)
⇒ \(\vec r = (-5\hat i + 7\hat j - 3\hat k)+ \lambda (3\hat i + 2\hat j + 2\hat k)\)
∴ The vector equation for the line is \(\vec r = (-5\hat i + 7\hat j - 3\hat k)+ \lambda (3\hat i + 2\hat j + 2\hat k)\)
Last updated on May 26, 2025
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