Show that the three lines with direction cosines\(\frac{12}{13}\),\(-\frac{3}{13}\),\(-\frac{4}{13}\) ; \(\frac{4}{13}\),\(\frac{12}{13}\),\(\frac{3}{13}\);\(\frac{3}{13}\),\(-\frac{4}{13}\),\(\frac{12}{13}\) are mutually perpendicular.
Find the angle between the following pairs of lines:
(i)\(\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}\) and \(\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}\)
(ii \(\frac{x}{2}=\frac{y}{2}=\frac{z}{1}\) and \(\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}\)
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
(a) 2x+3y+4z-12=0 (b) 3y+4z-6=0
(c)x+y+z=1 (d) 5y+8=0
Find the vector and cartesian equation of the planes
(a) that passes through the point (1,0,-2)and the normal to the plane is \(\hat i+\hat j-\hat k\).
(b) that passes through the point(1,4,6)and the normal vector to the plane is \(\hat i-2\hat j+\hat k\).
If a line has the direction ratio -18,12and-4, then what are its direction cosines?
Find the direction cosines of a line which makes equal angle with the coordinate axes.
If a line makes angles 90°,135° and 45° with x,y and z-axes respectively, find its direction cosines.
If θ is the angle between any two vectors \(\vec a\) and \(\vec b\) , then|\(\vec a.\vec b\)|=|\(\vec a \times \vec b\)| when θ is equal to
Find the equation of the plane passing through the point (-1, 3, 2) and perpendicular to each of the planes x+2y+3z=5 and 3x+3y+z = 0.
Find the angles between the following pairs of lines:(i) \(\overrightarrow r= 2\hat i-5\hat j+\hat k+ \lambda (3\hat i-2\hat j+6\hat k)\)and
\(\overrightarrow r=7\hat i-6\hat k+\mu(\hat i+2\hat j+2\hat k)\)
(ii) \(\overrightarrow r=3\hat i+\hat j-2\hat k+\lambda(\hat i-\hat j-2\hat k)\) and
\(\overrightarrow r = 2\hat i-\hat j-56\hat k+\mu(3\hat i-5\hat j-4\hat k)\)
The value of \(\hat i\).(\(\hat j\)×\(\hat k\))+\(\hat j\).(\(\hat i\times\hat k\))+\(\hat k\).(\(\hat i\times \hat j\)) is
Find the scalar and vector components of the vector with initial point (2,1) and terminal point (-5,7).
Let \(\vec a\) anb \(\vec b\) be two unit vectors and θ is the angle between them.Then,\(\vec a+\vec b\) is a unit vector if
Find the values of x and y so that the vectors \(2 \hat {i} +3\hat{j}\) and \(x \hat {i} +y\hat{j}\) are equal.
If θ is the angle between two vectors \(\vec a\) and \(\vec b\), then \(\vec a.\vec b\)≥0 only when
Find the cartesian equations of the following planes:
\(\overrightarrow r.(\hat i+\hat j-\hat k)=2\) (b) \(\overrightarrow r.(2\hat i+3\hat j-4\hat k)=1\)
(c) \(\overrightarrow r.[(s-2t)\hat i+(3-t)\hat j+(2s+t)\hat k)=15\)
Prove that (\(\vec a+\vec b\)).(\(\vec a+\vec b\))=|\(\vec a\)|2+|\(\vec b\)|2, if and only if \(\vec a\),\(\vec b\) are perpendicular, given a≠0,b≠0.
Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector \(3\hat i+5\hat j-6\hat k.\)