Three Dimensional Geometry · Mathematics · Class 12

The angle which the line $\frac{x}{1}=\frac{y}{-1}=\frac{z}{0}$ makes with the positive direction of Y -axis is

The Cartesian equation of the line passing through the point $(1,-3,2)$ and parallel to the line $\vec{r}=(2+\lambda) \hat{i}+\lambda \hat{j}+(2 \lambda-1) \hat{k}$ is

If a line makes angles of $$90^{\circ}, 135^{\circ}$$ and $$45^{\circ}$$ with the $$x, y$$ and $$z$$ axes respectively, then its direction cosines are:

The angle between the lines $$2 x=3 y=-z$$ and $$6 x=-y=-4 z$$ is:

Find the vector equation of the line passing through the point $(2,3,-5)$ and making equal angles with the co-ordinate axes.

(a) Find the co-ordinates of the foot of the perpendicular drawn from the point $(2,3,-8)$ to the line $\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$.

OR

(b). Find the shortest distance between the lines $\mathrm{L}_1 \& \mathrm{~L}_2$ given below: $\mathrm{L}_1$ : The line passing through $(2,-1,1)$ and parallel to $\frac{x}{1}=\frac{y}{1}=\frac{z}{3} \mathrm{~L}_2: \vec{r}=\hat{i}+(2 \mu+1) \hat{j}-(\mu+2) \hat{k}$.

Find the vector and the cartesian equations of a line that passes through the point $$A(1,2,-1)$$ and parallel to the line $$5 x-25=14-7 y=35 z$$.

(a) Find the coordinates of the foot of the perpendicular drawn from the point $$P(0,2,3)$$ to the line $$\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$$.

OR

(b) Three vectors $$\vec{a} \cdot \vec{b}$$ and $$\vec{c}$$ satisfy the condition $$\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$$. Evaluate the quantity $$\mu= \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$$, if $$|\vec{a}|=3,|\vec{b}|=4$$ and $$|\vec{c}|=2$$