Find the coordinates of the point where the l | vedclass

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Question

(A) The vector equation of the line passing through points $A(3, 4, 1)$ and $B(5, 1, 6)$ is given by $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$.

Vedclass · Accepted Answer

$\left( \frac{13}{5}, \frac{23}{5}, 0 \right)$

Vedclass · Answer

(A) The vector equation of the line passing through points $A(3, 4, 1)$ and $B(5, 1, 6)$ is given by $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$.Substituting the coordinates,we get $\vec{r} = (3\hat{i} + 4\hat{j} + \hat{k}) + \lambda((5-3)\hat{i} + (1-4)\hat{j} + (6-1)\hat{k})$.This simplifies to $\vec{r} = (3 + 2\lambda)\hat{i} + (4 - 3\lambda)\hat{j} + (1 + 5\lambda)\hat{k}$.Since the line crosses the $XY$-plane,the $z$-coordinate of any point on the line must be $0$.Therefore,$1 + 5\lambda = 0$,which gives $\lambda = -\frac{1}{5}$.Substituting $\lambda = -\frac{1}{5}$ into the expressions for $x$ and $y$:$x = 3 + 2(-\frac{1}{5}) = 3 - \frac{2}{5} = \frac{13}{5}$.$y = 4 - 3(-\frac{1}{5}) = 4 + \frac{3}{5} = \frac{23}{5}$.Thus,the coordinates of the point are $\left( \frac{13}{5}, \frac{23}{5}, 0 \right)$.

Find the coordinates of the point where the line passing through the points $A(3, 4, 1)$ and $B(5, 1, 6)$ crosses the $XY$-plane.

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