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Question: The mosquito net over a \[7m\times 4m\] bed is \[3m\] high. The net has a hole at one corner of the ...

The mosquito net over a 7m×4m7m\times 4m bed is 3m3m high. The net has a hole at one corner of the bed through which a mosquito enters the net. It flies and sits at the diagonally opposite upper corner of the net.
(a) Find the magnitude of the displacement of the mosquito.
(b) Taking the hole as the horizon, the length of the bed as X-axis, its width as Y-axis and vertically up as the Z-axis, write the components of the displacement vector.

Explanation

Solution

The change in the position vector (the position of an object with respect to the origin) of an object is known as the displacement vector. To find the displacement vector, we need to find the final position of the object. Once we find that, we have to find the difference between the initial and the final position vector to find the displacement vector and the magnitude of the displacement vector will give us the part (a) of our answer.

Formula Used:
displacement vector = final position vector - initial position vector\text{displacement vector = final position vector - initial position vector}

Complete step by step solution:
Since the mosquito enters the net through the hole and the hole is taken as the horizon, or the origin, we can say that initial position vector of the mosquito is (0,0,0)(0,0,0)

The mosquito reaches the top opposite corner of the net. We can consider the movement of the mosquito in two parts, one when it reaches the end of the diagonal and then rises to reach the top corner.

When the mosquito reaches the end of the diagonal across the floor, the position vector of the mosquito can be given as (7,4,0)(7,4,0) since the length of the bed represents the x-axis and the width of the bed represents the y-axis.

When the mosquito rises to the top corner, the final position vector of the mosquito would be (7,4,3)(7,4,3) since the height above the bed falls under the z-axis.
Now that we have the initial and the final position vectors of the mosquito, the displacement vector can be given as d→=(7−0)i^+(4−0)j^+(3−0)z^\overrightarrow{d}=(7-0)\widehat{i}+(4-0)\widehat{j}+(3-0)\widehat{z}
The displacement vector (d→)= 7i^+4j^+3z^(\overrightarrow{d})=\text{ 7}\widehat{i}\text{+4}\widehat{j}\text{+3}\widehat{z}

The magnitude of the displacement of the vector (d)=72+42+32=8.06m(d)=\sqrt{{{7}^{2}}+{{4}^{2}}+{{3}^{2}}}=8.06m.

Note: If we had to find just the magnitude of displacement, we could have made use of the Pythagoras theorem to find the length of the diagonal across the floor of the net and then applied Pythagoras theorem on the obtained value of the diagonal and the height of the net to find the magnitude of displacement of the mosquito.