To find โ Angle between the two pair of lines Direction ratios of L1 = (-3,-2,0) Direction ratios of L2 = (1,-3,2) Tip โ If (a,b,c) be the direction ratios of the first line and (aโ,bโ,cโ) be that...
Find the coordinates of the foot of the perpendicular drawn from the point $(1,2,3)$ to the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{\mathrm{z}-7}{-2}$. Also, find the length of the...
Show that the lines $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{3}=z$ and $\frac{\mathrm{x}+1}{5}=\frac{\mathrm{y}-2}{1}, Z=2$ do not intersect each other. Answer Given: The equations of the two...
Find the vector equation of the line passing through the point with position vector $(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-5 \hat{\mathrm{k}})$ and parallel to the vector $(\hat{i}+3...
A line passes through the point $(2,1,-3)$ and is parallel to the vector $(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$. Find the equations of the line in vector and Cartesian forms....
Answer Given: line passes through point $(3,4,5)$ and is parallel to $2 \hat{\imath}+2 \hat{\jmath}-3 \hat{k}$ To find: equation of line in vector and Cartesian forms Formula Used: Equation of a...
90C88 Answer : We know that: โ90C88=4005 Ans: โ90C88=4005 ย
16c13 Answer : We know that: nCr = ย ย ย โ16C13= โ16C13= โ16C13 = โ16C13 = โ16C13 = โ16C13 =560 Ans: 16C13=560
21C4 Answer : We know that: nCr โ20C4 โ20C4 โ20C4 โ20C4 โ20C4 โ20C4 =4845 Ans: 20C4 =4845
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Answer : It is required to find the sum (41 + 42 + 43 + โฆ. + 100). (41 + 42 + 43 + โฆ. + 100) = Sum of integers starting from 1 to 100 โ Sum of integers starting from 1 to 40. Note:ย Sum of first n...
Answer : It is required to find the sum of (2 + 4 + 6 + 8 +โฆ 100).
Answer : In the given question we need to find the sum of the series. For that, first, we need to find the nth term of the series so that we can use summation of the series with standard...
Answer : In the given question we need to find the sum of the series. For that, first, we need to find the nth term of the series so that we can use summation of the series with standard identities...
with the x-axis, then find the value of ฮป.Answer: Direction ratios of the x-axis are 1,0,0 and direction ratios of normal to the plane are 2,โ3,โ6. The angle between the line and the plane,...
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and the plane 10x + 2y โ 11z = 3.Answer: Direction ratios of the given line are 2,3,6 Direction ratios of the normal to the given plane are 10,2,โ11 The angle between the line and the plane: \begin{array}{l} \sin...
and the plane 
Answer: The equation of line is: $\vec{r}=(i+2 \hat{j}-\hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$ Here, $\vec{b}=\hat{i}-j+\vec{k}$ The equation of plane is: r→·(2i^-j^+k^)=4 \vec{r}...
and the plane 
Answer: The equation of plane is: r→=(2i^-j^+3k^)+λ(3i^-j^+2k^) \vec{r}=(2 \hat{i}-\hat{j}+3 \hat{k})+\lambda(3 \hat{i}-\hat{j}+2 \hat{k}) On comparing with $\vec{r}=\vec{a}+\lambda...
is parallel to the plane 
Answer: Given equation of line, $\overrightarrow rย = (\widehat i + 2\widehat k) + \lambda (2\widehat i ย - m\widehat j - 3\widehat k)$ Comparing with the line $\overrightarrow rย = \overrightarrow...
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is parallel to the plane 
. Also, find the distance between them.Answer: A line $\begin{array}{l} \overrightarrow rย = \overrightarrow aย + \lambda \overrightarrow b \\ \end{array}$ is parallel to the plane $\begin{array}{l} \overrightarrow r .\overrightarrow nย ...
and the plane 
Answer: The equation of line is: r→=(3i^+k^)+λ(j^+k^) \vec{r}=(3 \hat{i}+\hat{k})+\lambda(\hat {j}+\hat{k}) Comparing with $\vec{r}=\vec{a}+\lambda \vec{b}$ $\vec{b}=(j+\dot{k})$. The...
![\[\frac{{x - 2}}{3} = \frac{{y + 1}}{{ - 1}} = \frac{{z - 3}}{{ - 2}}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-24eba3255120c3288d2d203f342ddf45_l3.png "Rendered by QuickLaTeX.com")
Answer: The given line is $\frac{x-2}{3}=\frac{y+1}{-1}=\frac{2-3}{2}$ ⇒r→=(2i→-j→+k→)+t(3i→-j→-2k→) \Rightarrow...
Solution: Option(C) To find: Value of $\left|\begin{array}{cc}a+i b & c+i d \\ -c+i d & a-i b\end{array}\right|$ Formula used: $\mathrm{i}^{2}=-1$ We have, $\left|\begin{array}{cc}a+i b...
Solution: Option(C) To find: Value of $\left|\begin{array}{ll}\cos 15^{\circ} & \sin 15^{\circ} \\ \sin 15^{\circ} & \cos 15^{\circ}\end{array}\right|$ Formula used: (i) $\cos (A+B)=\cos A...
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are coplanar. Also find the equation of the plane containing these lines.Answer: - 7x + 14 + 14y โ 56 โ 7z + 42 = 0 - 7x + 14y โ 7z = 0 Multiply by negative sign, x โ 2y + z = 0 The equation of...
are coplanar. Also find the equation of the plane containing these lines.Answer:
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lies in the plane.Answer: ...
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are coplanar. Also find the equation of the plane containing these lines.Answer:
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![\[\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{16}=1\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-ad127cbd28649d4a6ca9eaa984f44acb_l3.png "Rendered by QuickLaTeX.com")
Given Equation: \[\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{16}=1\] Comparing with the equation of hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] we get, a = 3 and b = 4 (i)...
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.Answer: (i) For plane perpendicular, cos 900 = 0 (ii) For plane perpendicular, cos 900 = 0 (ฮป ร 1) + (2 ร...
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and 
.Answer: (i) (ii) = 6 + 24 โ 30 = 30 โ 30 = 0 = RHS Hence, the planes are...
Answer: (i) The standard equation of plane, (ii) The standard equation of plane, ...
Answer: (i) The standard equation of plane, (ii) The standard equation of plane, ...
Answer: (i) if ฮธ = 900 then cos 900 = 0 A1A2 + B1B2 + C1C2 = 0 By comparing with the standard equation of a plane, A1 = 3, B1 = 4, C1 = -5 A2 = 2, B2 = 6, C2 = 6 LHS = A1A2 + B1B2 + C1C2 = (3 ร 2) +...
Answer: Cartesian form of equation of plane through the line of intersection of planes: ...
Answer: Cartesian form of equation of plane through the line of intersection of planes: The equation...
Answer: Cartesian form of equation of plane through the line of intersection of planes: - x โ 27y โ 13z = 0 Multiplying by negative sign, x + 27y + 13z = 0 The equation of the plane is x + 27y + 13z...
Answer: Cartesian form of equation of plane through the line of intersection of planes: The equation of the plane is 7x + 13y + 4z = 9.
Answer: Cartesian form of equation of plane through the line of intersection of planes: 14x + 14y + 14z โ 84 + 6x + 9y + 12z + 15 = 0 20x + 23y + 26z...
Answer: Any plane parallel to 2x โ y + 3z + 7 = 0 is shown as 2x โ y + 3z + d = 0. It passes through the point (1, 4, -2) $\begin{array}{l} 2 \times 1 - 4 + 3( - 2) + d = 0\\ 2 - 4 - 6 + d = 0\\ d =...
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and parallel to the plane 
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