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
In a box containing 100 bulbs,10 are defective. The probability that out of a sample of 5 bulbs, none is defective is:
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.
If \(\vec a\),\(\vec b\),\(\vec c\)are mutually perpendicular to equal magnitudes, showing that the vector \(\vec a\)+\(\vec b\)+\(\vec c\)is equally inclined to \(\vec a\),\(\vec b\),and \(\vec c\).
Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector \(3\hat i+5\hat j-6\hat k.\)
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.