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 \(\vec{b}\)?
Find the area of the smaller region bounded by the ellipse \(\frac{x^2}{a^2}\)+\(\frac{y^2}{b^2}\)=1 and the line \(\frac{x}{a}\)+\(\frac{y}{b}\)=1
Determine order and degree(if defined)of differential equation \(\frac{d^4y}{dx^4}\)+sin(y''')=0
\(If ƒ(x)=∫_0^xt\ sin\ t\ dt,\ then ƒ'(x)is\)
Find the area of the smaller region bounded by the ellipse \(\frac{x^2}{9}\)+\(\frac{y^2}{4}\)=1 and the line
\(\frac{x}{3}\)+\(\frac{y}{2}\)=1
Find the area enclosed by the parabola 4y=3x2 and the line 2y=3x+12