Find the coordinates of the foot of the perpendicular drawn from the point
Question: Find the coordinates of the foot of the perpendicular drawn from the point $A(1,8,4)$ to the line joining the points $B(0,-1,3)$ and $C(2,-3,-1)$. Solution: $R\left(-\frac{5}{3}, \frac{2}{3}, \frac{19}{3}\right)$...
Read More →Find the angles made by the following vectors with the coordinate axes:
Question: Find the angles made by the following vectors with the coordinate axes: (i) $(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})$ (ii) $(\hat{\mathrm{j}}-\hat{\mathrm{k}})$ (iii) $(\hat{i}-4 \hat{j}+8 \hat{k})$ Solution:...
Read More →Find the angle between the vectors
Question: Find the angle between the vectors $\overrightarrow{r_{1}}=(3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{r}_{2}}=(4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+7 \hat{\mathrm{k}})$ Solution:...
Read More →Find the angle between the lines whose direction ratios are:
Question: Find the angle between the lines whose direction ratios are: $1,1,2$ and $(\sqrt{3-1}),(-\sqrt{3-1}), 4$ Solution:...
Read More →Find the angle between the lines whose direction ratios are:
Question: Find the angle between the lines whose direction ratios are: $2,-3,4$ and $1,2,1$. Solution:...
Read More →Find the angle between the two lines whose direction ratios are:
Question: Find the angle between the two lines whose direction ratios are: $a, b, c$ and $(b-c),(c-a),(a-b)$ Solution:...
Read More →Find the angle between the two lines whose direction cosines are:
Question: Find the angle between the two lines whose direction cosines are: $\frac{2}{3}, \frac{-1}{3}, \frac{-2}{3}$ and $\frac{3}{7}, \frac{2}{7}, \frac{6}{7}$ Solution:...
Read More →Find the value of p for which the points
Question: Find the value of $p$ for which the points $A(-1,3,2), B(-4,2,-2)$, and $C(5,5, p)$ are collinear. Solution:...
Read More →Solve this following
Question: Show that the points $A(-2,4.7), B(3,-6 .-8)$ and $C(1,-2,-2)$ are collinear. Solution:...
Read More →Solve this following
Question: Show that the points $A(2,3,4), B(-1,-2,1)$ and $C(5,8,7)$ are collinear. Solution:...
Read More →If the line segment joining the points
Question: If the line segment joining the points $A(7, p, 2)$ and $B(q,-2,5)$ be parallel to the line segment joining the points $C(2,-3,5)$ and $D(-6,-15,11)$, find the values of $p$ and $q$. Solution:...
Read More →Show that the line segment joining the points
Question: Show that the line segment joining the points $A(1,2,3)$ and $B(4,5,7)$ is parallel to the segment joining the points $C(-4,3,-6)$ and $D(2,9,2)$. Solution:...
Read More →Solve this following
Question: If $O$ is the origin and $P(2,3,4)$ and $Q(1,-2,1)$ be any two points show that $O P \perp O Q$. Solution:...
Read More →Solve this following
Question: Find the value of $p$ for which the line through the points $A(3,5,-1)$ and $B(5, p, 0) 9$ is perpendicular to the line through the points $C(2,1,1)$ and $D(3,3,-1)$. Solution:...
Read More →Show that the line segment joining the origin to the point
Question: Show that the line segment joining the origin to the point $A(2,1,1)$ is perpendicular to the line segment joining the points $B(3,5,-1)$ and $C(4,3,-1)$. Solution:...
Read More →Show that the line joining the points
Question: Show that the line joining the points $A(1,-1,2)$ and $B(3,4,-2)$ is perpendicular to the line joining the points $C(0,3,2)$ and D $(3,5,6)$. Solution:...
Read More →Find the direction ratios and the direction cosines of the line segment joining the points:
Question: Find the direction ratios and the direction cosines of the line segment joining the points: (i) $A(1,0,0)$ and $B(0,1,1)$ (ii) $A(5,6,-3)$ and $B(1,-6,3)$ (iii) $A(-5,7,-9)$ and $B(-3,4,-6)$ Solution:...
Read More →Find the direction cosines of a line segment whose direction ratios are:
Question: Find the direction cosines of a line segment whose direction ratios are: (i) $2 .-6,3$ (ii) $2,-1,-2$, (iii) $-9,6,-2$ Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: For any three vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ the value of $[\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}} \overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}]$ is A. $2[\vec{a} \vec{b} \vec{c}]$ B. 1 C. 0 D. none of these Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $\vec{a} \cdot(\vec{a} \times \vec{b})=?$ A. 0 B. 1 C. $\mathbf{a}^{2} \mathbf{b}$ D. meaningless Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: Which of the following is meaningless? A. $\vec{a} \cdot(\vec{b} \times \vec{c})$ B. $\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}})$ C. $(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \cdot \overrightarrow{\mathrm{c}}$ D. none of these Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: It is given that the vectors $\overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}-2 \hat{\mathrm{k}}), \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+(\lambda+1) \hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{c}}=(4 \hat{\mathrm{i}}+2 \hat{\mathrm{k}})$ are coplanar. Then, the value of $\lambda$ is A. $\frac{1}{2}$ B. $\frac{1}{3}$ C. 2 D. 1 Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If the volume of a parallelepiped having $\vec{a}=(5 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}+\hat{\mathrm{k}}), \overrightarrow{\mathrm{b}}=(4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}})$ $\overrightarrow{\mathrm{c}}=(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+7 \hat{\mathrm{k}})$ as conterminous edges, is 216 cubic units then the value of $\lambda$ is A. $\frac{5}{3}$ B. $\frac{4}{3}$ C. $\frac{2}{3}$ D. $\fr...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: If $\overrightarrow{\mathrm{a}}=(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}), \overrightarrow{\mathrm{b}}=(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{c}}=(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}})$ be the coterminous edges of a parallelepiped then its volume is A. 21 cubic units B. 14 cubic units C. 7 cubic units D. none of these Solution:...
Read More →Mark against the correct answer in each of the following:
Question: Mark $(\sqrt{)}$ against the correct answer in each of the following: $[\hat{\mathrm{i}} \hat{\mathrm{j}} \hat{\mathrm{k}}]=?$ A. 0 B. 1 C. 2 D. 3 Solution:...
Read More →