Find the general solution of the differential equation:\(\frac {dy}{dx}+\sqrt {\frac {1-y^2}{1-x^2}}=0\)
In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin. (a) z=2 (b) x+y+z=1 (c)2x+3y-z=5 (d) 5y+8=0
Form the differential equation representing the family of curves given by:\((x-α)^2+2y^2=α^2\) where a is an arbitrary constant.
If \(α→s\) a nonzero vector of magnitude \('α'\) and \(λ\) a nonzero scalar,then \(λ\vec{α}\) is unit vector if
Show that the lines\(\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1}\) and \(\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\) are perpendicular to each other.
Find the values of P so the line\(\frac{1-x}{3}=\frac{7y-14}{2p}=\frac{z-3}{2}\) and \(\frac{7-7x}{3p}=\frac{y-5}{1}=\frac{6-z}{5}\) are at right angles.
Show that the vectors \(2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}\) and \(3\hat{i}-4\hat{j}-4\hat{k}\) from the vertices of a right-angled triangle.
The scalar product of the vector \(\hat i+\hat j+\hat k \) with a unit vector along the sum of vectors \(2\hat i+4\hat j-5 \hat k\) and \(\lambda \hat i+2\hat j+3\hat k\) is equal to one. Find the value of λ.
Find the coordinates of the point where the line through (3,-4,-5) and (2,-3,1)crosses the plane 2x+y+z=7.
Find the coordinates of the point where the line through (5,1,6) and (3,4,1) crosses the ZX-plane.
Find the coordinates of the point where the line through (5,1,6)and (3,4,1) crosses the YZ plane.
Find the shortest distance between lines \(\overrightarrow{r}\)=\(6\hat i+2\hat j+2\hat k\)+λ(\(\hat i+2\hat j+2\hat k\))and\(\overrightarrow{r}\)=-\(-4\hat i-\hat k\)+μ(\(3\hat i+2\hat j+2\hat k\)).
If the vertices \(A,B,C\) of a triangle \(ABC\) are\((1,2,3),(-1,0,0),(0,1,2)\), respectively,then find \(\angle{ABC}\).[\(\angle{ABC}\) is the triangle between the vectors\( \overrightarrow{BA}\)and \( \overrightarrow{BC}\)].
Find the equation of the plane passing through (a,b,c)and parallel to the plane \(\overrightarrow{r}\).(\(\hat i+\hat j+\hat k\))=2.
