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(a) What are the components of a vector a in the xy plane if its direction is 252° counterclockwise from the positive x axis and its magnitude is 7.34 units? (b) The x component of a certain vector is -25 units and the y component is +43 units. What are the magnitude of the vector and the angle between its direction and the positive x axis?

(a) What are the components of a vector a in the xy plane if its direction is 252° counterclockwise from the positive x axis and its magnitude is 7.34 units? (b) The x component of a certain vector is -25 units and the y component is +43 units. What are the magnitude of the vector and the angle between its direction and the positive x axis?

Grade:10

1 Answers

Jitender Pal
askIITians Faculty 365 Points
6 years ago
In spite the fact that we are given the angle, that vector\overrightarrow{a} subtends relative to the positive x axis, it would be appropriate to calcite the components by , using the angle relative to the negative x axis.
Let us assume that the angle subtended by vector\overrightarrow{a} relative [\overrightarrow{a}] to the negative axis be represented by . We assume that the magnitude of vector is represented by , the horizontal vector component of vector represented by and the vertical vector component represented by vector .

One can find the angle (see the figure below) as:





Also, from the figure above, it is clear that the horizontal component of vector is opposite to the direction of unit vector whereas the vertical component of the vector is opposite to the direction of unit vector .

Therefore the components of both the vectors have a negative sign.

Given:



The horizontal vector component of vector is:


Substituting the given value of and calculated value of , we have


Therefore the horizontal vector component of vector is .

The vertical vector component of vector is:


Substituting the given value of and calculated value of , we have


Therefore the vertical vector component of vector is .


Let us assume that the vector is represented as , such that the horizontal vector component of the vector is given as whereas the vertical vector component as . We also assume that the magnitude of vector is represented by .
Given:


The magnitude of vector is given as:


Substituting the given values, we have


Rounding off to two significant figures, we have

Therefore the magnitude of vector is .
Let us assume that the angle made by the vector with the positive axis, measured counterclockwise is given b .

Therefore the angle is given as:

Substituting the given values, we have



We assumed above that the angle is measured counterclockwise; however the negative sign in the angle clearly suggest that the angle subtended by the vector with respect to positive axis is measure clockwise.

Therefore the vector is from the positive axis measured clockwise.

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