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A ship sets out to sail to a point 124 km due north. An unexpected storm blows the ship to a point 72.6 Ian to the north and 31.4 km to the east of its starting point. How far, and in what direction, must it now sail to reach its original destination?

A ship sets out to sail to a point 124 km due north. An unexpected storm blows the ship to a point 72.6 Ian to the north and 31.4 km to the east of its starting point. How far, and in what direction, must it now sail to reach its original destination?

Grade:11

1 Answers

Aditi Chauhan
askIITians Faculty 396 Points
6 years ago
Consider two vectors given as:
236-1575_1.PNG
Then the sum (say \overrightarrow{c}) of the above two vectors can be stated in unit vector notation as:
236-497_2.PNG
The magnitude of vector\overrightarrow{c} is:
236-1861_3.PNG
The angle (say \phi) made by the vector c with the horizontal is given as:
236-101_4.PNG
Given:
236-2110_5.PNG
(a) From above it is clear that the sum of the vectors\overrightarrow{a} and vector \overrightarrow{b} can be given in the unit vector notation as:
\overrightarrow{c} = \overrightarrow{a} + \overrightarrow{b}
Comparing the given vectors to the vectors represented by equation (1) and (2), we have
236-381_6.PNG
Therefore the sum of the vectors\overrightarrow{a} and \overrightarrow{b} is given by equation (3) as:
236-2327_7.PNG
Therefore the sum of the vectors in unit vector notation is2\widehat{i} + 5\widehat{j} .
(b)The magnitude of \overrightarrow{a}+\overrightarrow{b} can be calculated using equation (4) as:


236-1514_8.PNG

Substituting the values of component of vector \overrightarrow{a} and vector\overrightarrow{b} , we have
236-1758_9.PNG
Therefore the magnitude of sum of vectors \overrightarrow{a}and \overrightarrow{b}is \sqrt{29}.
The direction of vector \overrightarrow{c}can be calculated using equation (5) as:
236-1618_10.PNG
Substituting the values of component of vectors\overrightarrow{a} and , \overrightarrow{b}we have
236-401_11.PNG
Therefore the vector\overrightarrow{c} subtends an angle of 68.1° with the horizontal.

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