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Vector a has a magnitude of 5.2 units and is directed east. Vector b has a magnitude of 4.3 units and is directed 35° west of north. By constructing vector diagrams, find the magnitudes and directions of (a) a + b, and (b) a-b.

Vector a has a magnitude of 5.2 units and is directed east. Vector b has a magnitude of 4.3 units and is directed 35° west of north. By constructing vector diagrams, find the magnitudes and directions of (a) a + b, and (b) a-b.

Grade:10

2 Answers

Jitender Pal
askIITians Faculty 365 Points
6 years ago
The figure below shows the vector diagram, representing\overrightarrow{a} vectors and \overrightarrow{b}.
235-1546_chapter 14.png
To add the vectors, we need to resolve vector\overrightarrow{b} into its components along the unit vectors\widehat{i} and\widehat{j} respectively. The unit vector \widehat{j}points towards the north while the unit vector\widehat{i} points toward the east direction (as can be seen from the figure above).
We assume that the angle subtended by vector\overrightarrow{b} is\phi such that vector\overrightarrow{b} is\phi degrees west of north. We also assume that the magnitude of vector\overrightarrow{b} be represented by b whereas the magnitude of vector\overrightarrow{a} is represented by a .
Therefore the vector component of vector\overrightarrow{b} towards the north, along the unit vector \overrightarrow{j}\left ( say\ b_{x}\widehat{j} \right ) is given as whereas the vector component along the west, opposite to the direction of unit vector \widehat{i} (say\ b_{y}\widehat{i})is -bsin\ \phi \widehat{i}(refer figure below).

235-2288_Capture.PNG
Given:
a = 5.2 units
b = 4.3 units
\o = 35^{0}
Therefore one can write the vector\overrightarrow{a} as:
\overrightarrow{a} = a_{x} \widehat{i}
Where is the component of vector\overrightarrow{a} along the unit vector\widehat{i} .
Since vector \overrightarrow{a}points in the east, along the unit vector \widehat{i}, the value of componenta_{x} is equal to the magnitude of vector that is a .
Substituting the given values, we have
\overrightarrow{a} = 5.2 units\ \widehat{i} …… (1)
The vector is given as:
\overrightarrow{a} = b_{x} \widehat{i} + b_{y} \widehat{j}
\overrightarrow{b} = -b\ sin\ \o \widehat{i} + b\ cos\o \widehat{j}
\overrightarrow{b} = b (-sin\ \o\ \widehat{i} + cos\ \o\ \widehat{j})
Substituting the given values, we have
\overrightarrow{b} = 4.3 units (-sin35^{\circ}\widehat{j})
\overrightarrow{b} = -2.5 units\ \widehat{i}\ + 3.5\ units\ \widehat{j} …… (2)
The negative sign in the componentb_{x} of vector\overrightarrow{b} indicates that the component points opposite to the direction of unit vector\widehat{i} that is along west.
(a) The addition of vector\overrightarrow{a} and vector\overrightarrow{b} is given as:
\overrightarrow{a} + \overrightarrow{b} = (a_{x} + b_{x})\widehat{i} + (a_{y} + b_{y}) \widehat{j} …… (3)
Since there is no component of vector\overrightarrow{a} along the unit vector\widehat{j} , we have .
a_{y} = 0/ units.
From equations (1) and (2), we have
a_{x} = 5.2 units
b_{x} = -2.5\ units
b_{y} = 3.5 units
Substituting the above values in equation (3), we have
\overrightarrow{a} + \overrightarrow{b} = (5.2 units + (-2.5 units) \widehat{i} + (0\ units + 3.5\ units) \widehat{j}
= 2.7\ units\ \widehat{i} + 3.5\ units\ \widehat{j}
Therefore the vector is . \overrightarrow{a} + \overrightarrow{b} is\ 2.7 units\ \widehat{i}\ + 3.5\ units \widehat{j}.


Let us assume that the vector makes an angle with the horizontal. One can calculate the value of using the relation given below as:



Substituting the value of , , and from above, we have


The positive angle shows that the angle is measured counterclockwise with respect to horizontal.

Therefore vector makes an angle of with the horizontal.


The difference of vector and vector is given as:
…… (4)

Since there is no component of vector along the unit vector , we have .

From equations (1) and (2), we have



Substituting the given values in equation (4), we have



Therefore the vector is .


Let us assume that the vector makes an angle with the horizontal. One can calculate the value of using the relation given below as:







Substituting the value of , , and from above, we have


The negative sign above shows that the angle is measured clockwise with respect to the horizontal.

Therefore vector makes an angle of with the horizontal.
Saurabh Kumar
askIITians Faculty 2411 Points
6 years ago
In such a problems, always proceed like as
1. Draw east, west and north-south on horizontal and vertical direction.
2. Resolve the vectors along horizontal and vertical direction.
3. According to the requirement of question, find the resultant .

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