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.
Find the vector equation of the plane passing through (1,2,3)and perpendicular to the plane r→.(i^+2j^-5k^)+9=0.
Show that the direction cosines of a vector equally inclined to the axes OX, OY, and OZ are \(\frac{1}{\sqrt 3}\),\(\frac{1}{\sqrt 3}\),\(\frac{1}{\sqrt 3}\).
The two adjacent sides of a parallelogram are 2\(\hat{i}\)-4\(\hat j\)+5\(\hat k\)and \(\hat{i}\)-2\(\hat j\)-3\(\hat k\). Find the unit vector parallel to its diagonal. Also, find its area.
Find the equation of the tangent line to the curve \(y = x^2 − 2x + 7\) which is:(a) parallel to the line \(2x − y + 9 = 0 \)(b) perpendicular to the line \(5y − 15x = 13\)
If either vector \(\vec {a}=\vec{0}\space or\space \vec{b}=\vec{0}\), then \(\vec{a}.\vec{b}=0\).But the converse need not be true.Justify your answer with an example.
Find the position vector of point R which divides the line joining two points P and Q whose position vector is (2\(\vec a\)+\(\vec b\))and(\(\vec a\)-3\(\vec b\))externally in the ratio 1:2. Also, show that P is the midpoint of the line segment RQ.
If \(\vec{a}.\vec{a}=0\) and \(\vec{a}.\vec{b}=0,\)then what can be concluded about the vector \(\vec{b}\)?