Solve this following
Question: If $\left[\begin{array}{lll}1 \mathrm{x} 1\end{array}\right]\left[\begin{array}{ccc}1 2 3 \\ 4 5 6 \\ 3 2 5\end{array}\right]\left[\begin{array}{c}1 \\ -2 \\ 3\end{array}\right]=\mathrm{O}$, find $\mathrm{x}$ Solution:...
Read More →Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0
Question: Find the equation of the plane which is perpendicular to the plane 5x+ 3y+ 6z+ 8 = 0 and which contains the line of intersection of the planesx+ 2y+ 3z 4 = 0 and 2x+yz+ 5 = 0. Solution: The given planes are P1: 5x + 3y + 6z + 8 = 0 P2: x + 2y + 3z 4 = 0 P3: 2x + y z + 5 = 0 Now, the equation of the plane passing through the line of intersection of P1and P3is (x + 2y + 3z 4) + (2x + y z + 5) = 0 (1 + 2)x + (2 + )y + (3 )z 4 + 5 = 0 . (i) From the question its understood that plane (i) i...
Read More →Find the shortest distance between the lines given by
Question: Find the shortest distance between the lines given by $\vec{r}=(8+3 \lambda \hat{i}-(9+16 \lambda) \hat{j}+(10+7 \lambda) \hat{k}$ and $\vec{r}=15 \hat{i}+29 \hat{j}+5 \hat{k}+\mu(3 \hat{i}+8 \hat{j}-5 \hat{k})$ Solution: Given equations of lines are $\ddot{r}=(8+3 \lambda) \hat{i}-(9+16 \lambda) \hat{j}+(10+7 \lambda) \hat{k}$ ......(i) and $\quad \vec{r}=15 \hat{i}+29 \hat{j}+5 \hat{k}+\mu(3 \hat{i}+8 \hat{j}-5 \hat{k})$ .............(ii) Equation (i) can be re-written as $\vec{r}=8 ...
Read More →Solve this following
Question: If $A=\left[\begin{array}{cc}\cos \alpha \sin \alpha \\ -\sin \alpha \cos \alpha\end{array}\right]$, show that $A^{2}=\left[\begin{array}{cc}\cos 2 \alpha \sin 2 \alpha \\ -\sin 2 \alpha \cos 2 \alpha\end{array}\right]$ Solution:...
Read More →Solve this following
Question: If $F(x)=\left[\begin{array}{ccc}\cos x -\sin x 0 \\ \sin x \cos x 0 \\ 0 0 1\end{array}\right]$, show that $F(x) . F(y)=F(x+y)$ 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_{1...
Read More →Find the equation of the plane through the points (2, 1, –1)
Question: Find the equation of the plane through the points (2, 1, 1) and (1, 3, 4) and perpendicular to the planex 2y+ 4z= 10. Solution: We know that, equation of the plane passing through two points (x1, y1, z1) and (x2, y2, z2) with its normals direction ratios is a(x x1) + b(y y1) + c(z z1) = 0 . (i) Now, if the plane is passing through two points (2, 1, -1) and (-1, 3, 4) then a(x2-x1) + b(y2 y1) + c(z2 z1) = 0 a(-1 2) + b(3 1) + c(4 + 1) = 0 -3a + 2b + 5c = 0 . (ii) As the required plane i...
Read More →Find the equations of the line passing through the point (3,0,1)
Question: Find the equations of the line passing through the point (3,0,1) and parallel to the planesx+ 2y= 0 and 3yz= 0. Solution: Given point is (3, 0, 1) and the equation of planes are x + 2y = 0 . (i) and 3y z = 0 . (ii) Equation of any line l passing through (3, 0, 1) is l: (x 3)/a = (y 0)/b = (z 1)/c Now, the direction ratios of the normal to plane (i) and plane (ii) are (1, 2, 0) and (0, 3, 1). As the line is parallel to both the planes, we have 1.a + 2.b + 0.c = 0 ⇒ a + 2b + 0c = 0 and 0...
Read More →Find the length and the foot of perpendicular
Question: Find the length and the foot of perpendicular from the point (1, 3/2, 2) to the plane 2x 2y+ 4z+ 5 = 0. Solution: Given plane is 2x 2y + 4z + 5 = 0 and point (1, 3/2, 2) The direction ratios of the normal to the plane are 2, -2, 4 So, the equation of the line passing through (1, 3/2, 2) and direction ratios are equal to the direction ratios of the normal to the plane i.e. 2, -2, 4 is $\frac{x-1}{2}=\frac{y-\frac{3}{2}}{-2}=\frac{z-2}{4}=\lambda$ Now, any point in the plane is 2 + 1, -2...
Read More →Find the distance of a point (2, 4, –1) from the line
Question: Find the distance of a point (2, 4, 1) from the line $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$ Solution:...
Read More →Find the foot of perpendicular from
Question: Find the foot of perpendicular from the point $(2,3,-8)$ to the line $\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$. Also, find the perpendicular distance from the given point to the line. Solution: Given, Now, the coordinates of any point Q on the line are x = -2 + 4, y = 6 and z = -3 + 1 and the given point is P(2, 3, -8) The direction ratios of PQ are -2 + 4 2, 6 3, -3 + 1 + 8 i.e. -2 + 2, 6 3, -3 + 9 And the direction ratios of the given line are -2, 6, -3. If PQ line, then -2(-2 + 2) +...
Read More →Two systems of rectangular axis have the same origin.
Question: Two systems of rectangular axis have the same origin. If a plane cuts them at distancesa,b,canda,b,c, respectively, from the origin, prove that $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}$ Solution: Lets take OX, OY, OZ and ox, oy, oz to be two rectangular systems. And, the equations of two planes are...
Read More →Solve this
Question: If $A=\left[\begin{array}{ll}1 -1 \\ 2 -1\end{array}\right], B=\left[\begin{array}{ll}a -1 \\ b -1\end{array}\right]$ and $(A+B)^{2}=\left(A^{2}+B^{2}\right)$ then find the values of $a$ and $b$. Solution:...
Read More →O is the origin and A is (a, b, c). Find the direction cosines of
Question: O is the origin and A is (a,b,c). Find the direction cosines of the line OA and the equation of plane through A at right angle to OA. Solution: Given, O (0, 0, 0) and A(a, b, c) So, the direction ratios of OA = a 0, b 0, c 0 = a, b, c And, the direction cosines of line OA Now, the direction ratios of the normal to the plane are (a, b, c). We know that, the equation of the plan passing through the point A(a, b, c) is a(x a) + b(y b) + c(z c) = 0 ax a2+ by b2+ cz c2= 0 ax + by + cz = a2+...
Read More →If a variable line in two adjacent positions has direction cosines l,
Question: If a variable line in two adjacent positions has direction cosinesl,m,nandl+ dl,m+ dm,n+ dn, show that the small angle dq between the two positions is given by dq2= dl2+dm2+ dn2 Solution: Given that l, m, n andl+ dl,m+ dm,n+ dnare the direction cosines of a variable line in two positions l2+ m2+ n2= 1 .. (i) and (l+ dl)2+ (m+ dm)2+ (n+ dn)2=1 . (ii)...
Read More →Find the matrix A such that A
Question: Find the matrix A such that A. $\left[\begin{array}{ll}2 3 \\ 4 5\end{array}\right]=\left[\begin{array}{cc}0 -4 \\ 10 3\end{array}\right]$. Solution: Formula used :...
Read More →Find the angle between the lines whose direction cosines
Question: Find the angle between the lines whose direction cosines are given by the equationsl+m+n= 0,l2+m2n2= 0. Solution: Given equations are, l+m+n= 0 .. (i) l2+m2n2= 0 . (ii) From equation (i), we have n = (l + m) Putting the value of n is equation (ii), we get l2+ m2+ [-(l + m)]2= 0 l2+ m2 l2 m2 2lm = 0 -2lm = 0 lm = 0 ⇒ (- m n)m = 0 [Since, l = m n] (m + n)m = 0 ⇒ m = 0 or m = -n ⇒ l = 0 or l = -n Now, the direction cosines of the two lines are 0, -n, n and -n, 0, n ⇒ 0, -1, 1 and -1, 0, 1...
Read More →Find the equations of the two lines through the origin
Question: Find the equations of the two lines through the origin which intersect the line $\frac{x-3}{2}=\frac{y-3}{1}=\frac{z}{1}$ at angles of $\pi / 3$ each. Solution: Any point in the given line is x 3/ 2 = y 3/1 = z/1 = x = 2 + 3, y = + 3 and z = Let it be the coordinates of P So, the direction ratios of OP are (2 + 3 0), ( + 3 0) and ( 0) ⇒ 2 + 3, + 3, But the direction ratios of the line PQ are 2, 1, 1 Now, we know that 2+ 3 + 3 = 42+ 9 + 12 (On squaring on both sides) 32+ 9 + 6 = 0 2+ 3 ...
Read More →Find the matrix A such that
Question: Find the matrix A such that $\left[\begin{array}{cc}5 -7 \\ -2 3\end{array}\right], A=\left[\begin{array}{cc}-16 -6 \\ 7 2\end{array}\right]$. Solution: $A=\left[\begin{array}{ll}1 -4 \\ 3 -6\end{array}\right]$...
Read More →Find the equation of the plane through the points (2, 1, 0),
Question: Find the equation of the plane through the points (2, 1, 0), (3, 2, 2) and (3, 1, 7). Solution: Given points are (2, 1, 0), (3, 2, 2) and (3, 1, 7) As the equation of the plane passing through the points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is (x 2) (-21) (y 1)(7 + 2) + z (3) = 0 -21 (x 2) 9(y 1) + 3z = 0 -21x + 42 9y + 9 + 3z = 0 -21x 9y + 3z + 51 = 0 ⇒ 7x + 3y z 17 = 0 Thus, the required equation of plane is 7x + 3y z 17 = 0....
Read More →If the line drawn from the point (–2, – 1, – 3)
Question: If the line drawn from the point (2, 1, 3) meets a plane at right angle at the point (1, 3, 3), find the equation of the plane. Solution: Given, points (2, 1, 3) and (1, 3, 3) Direction ratios of the normal to the plane are (1 + 2, -3 + 1, 3 + 3) = (3, -2, 6) Now, the equation of plane passing through one point (x1, y1, z1) is a(x x1) + b(y y1) + c(z z1) = 0 3(x 1) 2(y + 3) + 6(z 3) = 0 3x 3 2y 6 + 6z 18 = 0 3x 2y + 6z 27 = 0 ⇒ 3x 2y + 6z = 27 Thus, the required equation of plane is 3x...
Read More →Find the equation of a plane which is at
Question: Find the equation of a plane which is at a distance 33 units from origin and the normal to which is equally inclined to coordinate axis. Solution: As the normal to the plane is equally inclined to the axes we have, cos = cos = cos So, cos2 + cos2 + cos2 = 1 3 cos2 = 1 ⇒ cos = 1/3 And, cos = cos = cos = 1/3 Thus, the equation of the plane is x + y + z = 9...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ll}3 2 \\ 1 1\end{array}\right]$, find the value of a and b such that $A^{2}+a A+b l=0$ Solution:...
Read More →Find the equation of a plane which bisects perpendicularly
Question: Find the equation of a plane which bisects perpendicularly the line joining the points A (2, 3, 4) and B (4, 5, 8) at right angles. Solution: Given coordinates are A (2, 3, 4) and B (4, 5, 8) Now, the coordinates of the mid-point C are (2+4/2, 3+5/2, 4+8/2) = (3, 4, 6) And, the direction ratios of the normal to the plane = direction ratios of AB = 4 2, 5 3, 8 4 = (2, 2, 4) Equation of the plane is a(x x1) + b(y y1) + c(z z1) = 0 2(x 3) + 2(y 4) + 4(z 6) = 0 2x 6 + 2y 8 + 4z 24 = 0 2x +...
Read More →Solve this following
Question: If $A=\left[\begin{array}{ll}3 1 \\ 7 5\end{array}\right]$, find $x$ and $y$ such that $A^{2}+x \mid=y A$ Solution:...
Read More →Solve for x and y, when
Question: Solve for $x$ and $y$, when $\left[\begin{array}{cc}3 -4 \\ 1 2\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]=\left[\begin{array}{l}3 \\ 11\end{array}\right]$ Solution:...
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