Find the values of x and y when
Question: Find the values of $x$ and $y$, when $\left[\begin{array}{cc}2 -3 \\ 1 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}1 \\ 3\end{array}\right]$ Solution: Given : $\left[\begin{array}{cc}2 -3 \\ 1 1\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}1 \\ 3\end{array}\right]$ To find: $x$ and $y$ Formula used :...
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Question: If $A=\left[\begin{array}{cc}1 2 \\ 4 -3\end{array}\right]$ and $f(x)=2 x^{3}+4 x+5$, find $f(A)$ Solution: Formula used...
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Question: If $A=\left[\begin{array}{cc}-1 2 \\ 3 1\end{array}\right]$, find $f(A)$, where $f(x)=x^{2}-2 x+3$ Solution: $f(A)=A^{2}-2 A+31=\left[\begin{array}{cc}12 -4 \\ -6 8\end{array}\right]$...
Read More →Prove that the lines x = py + q,
Question: Prove that the linesx=py+q,z=ry+sandx=py+q,z=ry+s are perpendicular ifpp +rr + 1 = 0. Solution:...
Read More →Prove that the line through A (0, –1, –1) and B (4, 5, 1)
Question: Prove that the line through A (0, 1, 1) and B (4, 5, 1) intersects the line through C (3, 9, 4) and D ( 4, 4, 4). Solution: Given points, A (0, 1, 1) and B (4, 5, 1) C (3, 9, 4) and D ( 4, 4, 4). Cartesian form of equation AB is...
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Question: If $A=\left[\begin{array}{ll}3 -2 \\ 4 -2\end{array}\right]$, find $k$ so that $A^{2}=k A-21$ Solution:...
Read More →Find the angle between the lines
Question: Find the angle between the lines $\vec{r}=3 \hat{i}-2 \hat{j}+6 \hat{k}+\lambda(2 \hat{i}+\hat{j}+2 \hat{k})$ and $\vec{r}=(2 \hat{j}-5 \hat{k})+\mu(6 \hat{i}+3 \hat{j}+2 \hat{k})$ Solution:...
Read More →Show that the matrix
Question: Show that the matrix $A=\left[\begin{array}{ll}2 3 \\ 1 2\end{array}\right]$ satisfies the equation $A^{3}-4 A^{2}+A=O$. Solution:...
Read More →Show that the lines
Question: Show that the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ intersect. Also, find their point of intersection. Solution: Given equation are,...
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Question: If $A=\left[\begin{array}{cc}3 1 \\ -1 2\end{array}\right]$, show that $(A 2-5 A+7 I)=0$ Solution:...
Read More →Find the vector equation of the line which is parallel
Question: Find the vector equation of the line which is parallel to the vectorand which passes through the point (1, 2, 3). Solution: We know that the equation of line is...
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Question: If $A=\left[\begin{array}{cc}2 -2 \\ -3 4\end{array}\right]$ then find $\left(-A^{2}+6 A\right)$ Solution:...
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Question: If $A=\left[\begin{array}{cc}2 -1 \\ 3 2\end{array}\right]$ and $B=\left[\begin{array}{cc}0 4 \\ -1 7\end{array}\right]$, find $\left(3 A^{2}-2 B+I\right)$ Solution: Formula used :...
Read More →Find the position vector of a point A in space such
Question: Find the position vector of a point A in space such thatis inclined at 60 to OX and at 45 to OY and= 10 units. Solution: We know that, cos2 + cos2 + cos2 = 1 cos260o+ cos245o+ cos2 = 1 (1/2)2+ (1/2)2+ cos2 = 1 + + cos2 = 1 cos2 = 1 = So, cos = ⇒ cos = [Rejecting cos = , as 90o] Now, $\overrightarrow{\mathrm{OA}}=|\overrightarrow{\mathrm{OA}}|\left(\frac{1}{2} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}+\frac{1}{2} \hat{k}\right)=10\left(\frac{1}{2} \hat{i}+\frac{1}{\sqrt{2}} \hat{j}+\frac{1}{2}...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ccc}4 -1 -4 \\ 3 0 -4 \\ 3 -1 -3\end{array}\right]$, show that $A^{2}=1$ Solution: $\left[\begin{array}{lll}a_{11} a_{12} a_{13} \\ a_{21} a_{22} a_{23} \\ a_{31} a_{32} a_{33}\end{array}\right] \times\left[\begin{array}{lll}b_{11} b_{12} b_{13} \\ b_{21} b_{22} b_{23} \\ b_{31} b_{32} b_{33}\end{array}\right]$ $=\left[\begin{array}{lll}a_{11} b_{11}+a_{12} b_{21}+a_{13} b_{31} a_{11} b_{12}+a_{12} b_{22}+a_{13} b_{32} a_{11} b_{13}+a_{12} b_{23}+a_{13} b_{33}...
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Question: If $\mathrm{A}=\left[\begin{array}{ccc}2 -2 -4 \\ -1 3 4 \\ 1 -2 -3\end{array}\right]$, show that $A^{2}=A$ Solution: $\left[\begin{array}{lll}a_{11} a_{12} a_{13} \\ a_{21} a_{22} a_{23} \\ a_{31} a_{32} a_{33}\end{array}\right] \times\left[\begin{array}{lll}b_{11} b_{12} b_{13} \\ b_{21} b_{22} b_{23} \\ b_{31} b_{32} b_{33}\end{array}\right]$ $=\left[\begin{array}{lll}a_{11} b_{11}+a_{12} b_{21}+a_{13} b_{31} a_{11} b_{12}+a_{12} b_{22}+a_{13} b_{32} a_{11} b_{13}+a_{12} b_{23}+a_{1...
Read More →The angle between two vectors
Question: The angle between two vectors $\vec{a}$ and $\vec{b}$ with magnitudes $\backslash 3$ and 4 , respectively, and $\vec{a} \cdot \vec{b}=2 \sqrt{3}$ is (A) $\frac{\pi}{6}$ (B) $\frac{\pi}{3}$ (C) $\frac{\pi}{2}$ (D) $\frac{5 \pi}{2}$ Solution: The correct option is (B)....
Read More →The vector having initial and terminal
Question: The vector having initial and terminal points as (2, 5, 0) and (3, 7, 4), respectively is (A) $-\hat{i}+12 \hat{j}+4 \hat{k}$ (B) $5 \hat{i}+2 \hat{j}-4 \hat{k}$ (C)$-5 \hat{i}+2 \hat{j}+4 \hat{k}$ (D)$\hat{i}+\hat{j}+\hat{k}$ Solution: The correct option is (C). Let A and B be two points whose coordinates are given as (2, 5, 0) and (-3, 7, 4) So, we have...
Read More →Solve this following
Question: If $A=\left[\begin{array}{cc}a b b^{2} \\ -a^{2} -a b\end{array}\right]$, show that $A^{2}=0$ Solution:...
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Question: If $\mathrm{A}=\left[\begin{array}{ccc}1 0 -2 \\ 3 -1 0 \\ -2 1 1\end{array}\right], B=\left[\begin{array}{ccc}0 5 -4 \\ -2 1 3 \\ 1 0 2\end{array}\right]$ and $C=\left[\begin{array}{ccc}1 5 2 \\ -1 1 0 \\ 0 -1 1\end{array}\right]$; verify that $A(B-C)=(A B-A C)$ Solution: $\left[\begin{array}{lll}a_{11} a_{12} a_{13} \\ a_{21} a_{22} a_{23} \\ a_{31} a_{32} a_{33}\end{array}\right] \times\left[\begin{array}{lll}b_{11} b_{12} b_{13} \\ b_{21} b_{22} b_{23} \\ b_{31} b_{32} b_{33}\end{a...
Read More →The position vector of the point which divides the
Question: The position vector of the point which divides the join of points $2 \vec{a}-3 \vec{b}$ and $\vec{a}+\vec{b}$ in the ratio $3: 1$ is (A) $\frac{3 \vec{a}-2 \vec{b}}{2}$ (B) $\frac{7 \vec{a}-8 \vec{b}}{4}$ (C) $\frac{3 \vec{a}}{4}$ (D) $\frac{5 \vec{a}}{4}$ Solution: The correct option is (D). The given vectors are in the ratio 3: 1 Sor the position vector of the required point $c$ which divides the join of the given vectors $\vec{a}$ and $\vec{b}$ is $\vec{c}=\frac{m_{1} x_{2}+m_{2} x_...
Read More →The vector in the direction of the vector
Question: The vector in the direction of the vector $\hat{i}-2 \hat{j}+2 \hat{k}$ that has magnitude 9 is (A) $\hat{i}-2 \hat{j}+2 \hat{k}$ (B) $\frac{\hat{i}-2 \hat{j}+2 \hat{k}}{3}$ (C) $3(\hat{i}-2 \hat{j}+2 \hat{k})$ (D) $9(\hat{i}-2 \hat{j}+2 \hat{k})$ Solution: The correct option is (C)....
Read More →Prove the following
Question: If $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k}$, find a vector $\vec{c}$ such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$. Solution:...
Read More →Show that area of the parallelogram whose diagonals are given by
Question: Show that area of the parallelogram whose diagonals are given byandis . Also find the area of the parallelogram whose diagonals areand. $\frac{|\vec{a} \times \vec{b}|}{2}$ Solution: Lets take ABCD to be a parallelogram such that...
Read More →The value of
Question: $\frac{1}{2}[\vec{b} \times \vec{c}+\vec{c} \times \vec{a}+\vec{a} \times \vec{b}]$ 16. If $\vec{a}, \vec{b}, \vec{c} \quad$ determine the vertices of a triangle, show that gives the vector area of the triangle. Hence deduce the condition that the three points $\vec{a}, \vec{b}, \vec{c}$ are collinear. Also find the unit vector normal to the plane of the triangle. Solution:...
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