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प्रश्न
The equation of the line passing through the points \[a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \text{ and } b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \] is
पर्याय
\[\overrightarrow{r} = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) + \lambda \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
\[\overrightarrow{r} = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) - t \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
\[\overrightarrow{r} = a_1 \left( 1 - t \right) \hat{i} + a_2 \left( 1 - t \right) \hat{j} + a_3 \left( 1 - t \right) \hat{k} + t \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
none of these
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उत्तर
\[\overrightarrow{r} = a_1 \left( 1 - t \right) \hat{i} + a_2 \left( 1 - t \right) \hat{j} + a_3 \left( 1 - t \right) \hat{k} + t \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
Equation of the line passing through the points having position vectors
\[a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \text{ and } b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \] is
\[\overrightarrow{r} = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) + t\left\{ \left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right) - \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) \right\}, \text{ where t is a parameter } \]
\[ = \left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) - t\left( a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \right) + t\left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
\[ = a_1 \left( 1 - t \right) \hat{i} + a_2 \left( 1 - t \right) \hat{j} + a_3 \left( 1 - t \right) \hat{k} + t\left( b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \right)\]
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Find the angle between the pairs of lines with direction ratios proportional to a, b, c and b − c, c − a, a − b.
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The straight line \[\frac{x - 3}{3} = \frac{y - 2}{1} = \frac{z - 1}{0}\] is
Show that the lines \[\frac{5 - x}{- 4} = \frac{y - 7}{4} = \frac{z + 3}{- 5} \text { and } \frac{x - 8}{7} = \frac{2y - 8}{2} = \frac{z - 5}{3}\] are coplanar.
Find the joint equation of pair of lines through the origin which is perpendicular to the lines represented by 5x2 + 2xy - 3y2 = 0
Find the cartesian equation of the line which passes ·through the point (– 2, 4, – 5) and parallel to the line given by.
`(x + 3)/3 = (y - 4)/5 = (z + 8)/6`
Find the vector equation of a line passing through a point with position vector `2hati - hatj + hatk` and parallel to the line joining the points `-hati + 4hatj + hatk` and `-hati + 2hatj + 2hatk`.
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