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Two vectors have equal magnitudes of 12.7 units. They are oriented as shown in Fig. and their vector sum is . Find (a) the x and)' components of . (b) the magnitude of , and (c) the angle makes with the +x axis

6 years ago

Let us assume that the vectors are given as:

We assume that the magnitude of vectors above are given as a , b and r respectively.

We also assume that the angle made by vector from positive x axis, measured anticlockwise is given by whereas the angle made by vector from positive x axis, measured counterclockwise is .

Given:

First we write the vectors and in unit vector notations so that we can calculate the vector using simple addition of their components

The figure below shows the vector diagram, representing vectors and respectively.

It can be seen from the figure that the angle (say ) made by vector from line AB, measured counterclockwise, is:

One should note that the above figure is not subjected to accurate scaling.

Therefore we can write the vector components of vector as:

The negative sign in the vector component is due to the fact that the vector component points in the direction opposite to that with the unit vector along the positive x axis.

Therefore vector is:

If the angle made by vector with respect to positive x axis, measured counterclockwise is , we can write the vector components of the vector as:

(a) The vector can now be given as:

Substituting the value of components of vectors and from equation (1) and (2), we have:

Now for 28.2°, = 46.8° , a = 12.7 units and b = 12.7 units, we can calculate the vector using equation (3). Since the magnitude of vectors and are equal, we can modify the equation (3) using a = b as:

One must note that we have just substituted the magnitude of vector with that of vector to obtain the equation above.

Now we substitute the given values of, = 28.2°, b = 12.7 units and calculated value of = 46.8° in equation (4) to obtain the vector as:

From calculated value of vector , we have

Therefore the horizontal component of vector is 2.54 units.

The vertical component of vector is 15.1 units.

(b) The magnitude of vector is:

Therefore the magnitude of vector is 15.3 units

(c) The angle made by vector with positive x axis, measured counterclockwise can be given as:

Therefore the angle subtended by vector is 80.4°with positive x axis, measured counterclockwise.

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