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Given two vectors, find the magnitudes and directions (with the + x axis ) of (a)

Given two vectors,  find the magnitudes and directions (with the + x axis ) of (a)

Grade:11

1 Answers

Aditi Chauhan
askIITians Faculty 396 Points
6 years ago

Let us assume that the vector is given as:
236-2036_1.PNG
Given:
236-1724_2.PNG

(a) The magnitude of vector is given as:

236-1881_3.PNG
On comparing equation (1) with the given vector , we have

ax = 4.0
ay = -3.0
Substituting the values of the components in equation (3), we have
236-1405_4.PNG
Therefore the magnitude of vector is .


Let us assume that the vector makes an angle with the positive axis. The sign of the angle will indicate if the angle is measured clockwise or counterclockwise from positive axis.

The angle , for vector , given by equation (1) is:






Since the value of the angle is negative, we have the angle subtended by the vector is measured clockwise from the positive axis.



The magnitude of vector is given as:
…… (4)

On comparing equation (2) with the given vector , we have


Substituting the values of the components in equation (4), we have



Therefore the magnitude of vector is .


Let us assume that the vector makes an angle with the positive axis. The sign of the angle will indicate if the angle is measured clockwise or counterclockwise from positive axis.

The angle , for vector , given by equation (2) is:


Since the value of the angle is positive, we have the angle subtended by the vector is measured counterclockwise from the positive axis
The vector is given as:
…… (5)

On comparing equation (1), (2) with the given vectors and , we have





Substituting the value of the components of vectors and in equation (5), we have:



We have written 5.0 as 5.00 to match the significant figures of the components.

Therefore the vector is .


The magnitude of vector can be deduced from equation (5) as:



Substituting the values of the components in equation above, we have



Rounding off to two significant figures, we have


Therefore the magnitude of vector is .


Let us assume that the vector makes an angle with the positive axis. The sign of the angle will indicate if the angle is measured clockwise or counterclockwise from positive axis.

The angle for vector , given by equation (5) is:


Since the value of the angle is positive, we have the angle subtended by the vector is measured counterclockwise from positive axis.


The vector is given as:
…… (6)

The vector components of vectors and are:





Substituting the value of the components of vector and vector in equation (6), we have:



We have written 2.0 as 2.00 to match the significant figures of the components.

Therefore the vector is .




The magnitude of vector can be deduced from equation (6) as:



Substituting the values of the components in equation above, we have



Rounding off to two significant figures, we have


Therefore the magnitude of vector is .


Let us assume that the vector makes an angle with the positive axis. The sign of the angle will indicate if the angle is measured clockwise or counterclockwise from positive axis.


The angle for vector , given by equation (6) is:


Since the value of the angle is positive, we have the angle subtended by the vector is measured counterclockwise from positive axis.


The vector is given as:
…… (7)

The vector components of vectors and are:





Substituting the value of the components of vector and vector in equation (7), we have:



We have written 2.0 as 2.00 to match the significant figures of the components.

Therefore the vector is .


The magnitude of vector can be deduced from equation (7) as:



Substituting the values of the components in equation above, we have



Rounding off to two significant figures, we have


Therefore the magnitude of vector is .


Let us assume that the vector makes an angle with the positive axis. The sign of the angle will indicate if the angle is measured clockwise or counterclockwise from positive axis.



The angle for vector using equation (7), can be given as:


Since the value of the angle is positive, we have the angle subtended by the vector is measured counterclockwise from positive axis.

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