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Show that the vector `hati + hatj + hatk` is equally inclined to the axes OX, OY, and OZ.
Concept: undefined >> undefined
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `hati + 2hatj - hatk` and `-hati + hatj + hatk` respectively, externally in the ratio 2:1.
Concept: undefined >> undefined
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Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).
Concept: undefined >> undefined
Show that the points A, B and C with position vectors `veca = 3hati - 4hatj - 4hatk`, `vecb = 2hati - hatj + hatk` and `vecc = hati - 3hatj - 5hatk`, respectively form the vertices of a right angled triangle.
Concept: undefined >> undefined
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of the x-axis.
Concept: undefined >> undefined
Find the value of x for which `x(hati + hatj + hatk)` is a unit vector.
Concept: undefined >> undefined
If θ is the angle between two vectors `veca` and `vecb`, then `veca . vecb >= 0` only when ______.
Concept: undefined >> undefined
Let `veca` and `vecb` be two unit vectors, and θ is the angle between them. Then `veca + vecb` is a unit vector if ______.
Concept: undefined >> undefined
Find the rate of change of the area of a circle with respect to its radius r when r = 3 cm.
Concept: undefined >> undefined
The volume of a cube is increasing at the rate of 8 cm3/s. How fast is the surface area increasing when the length of an edge is 12 cm?
Concept: undefined >> undefined
The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.
Concept: undefined >> undefined
An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?
Concept: undefined >> undefined
A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?
Concept: undefined >> undefined
The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?
Concept: undefined >> undefined
The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.
Concept: undefined >> undefined
A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.
Concept: undefined >> undefined
A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.
Concept: undefined >> undefined
A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?
Concept: undefined >> undefined
A particle moves along the curve 6y = x3 +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.
Concept: undefined >> undefined
The radius of an air bubble is increasing at the rate `1/2` cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?
Concept: undefined >> undefined
