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A car travels 6 km towards the north at an angle 45^ to the east and then travels distance of 4 km towards the north at an angle 135^ to the east. How far is the point from the starting point? What angle does the straight line joining its initial and final positions makes with the east?
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A car travels 6 km towards the north at an angle 45^ to the east and t...
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Problem:
A car travels 6 km towards the north at an angle 45^ to the east and then travels distance of 4 km towards the north at an angle 135^ to the east. How far is the point from the starting point? What angle does the straight line joining its initial and final positions makes with the east?

Solution:

Step 1: Draw a diagram
Draw a diagram and label the points as shown below.

![image.png](attachment:image.png)

Step 2: Find the horizontal and vertical components of the two vectors
Horizontal component of the first vector = 6*cos(45^) = 6/√2
Vertical component of the first vector = 6*sin(45^) = 6/√2

Horizontal component of the second vector = 4*cos(135^) = -4/√2
Vertical component of the second vector = 4*sin(135^) = 4/√2

Step 3: Add the horizontal and vertical components of the two vectors
Horizontal component of the resultant vector = (6/√2) + (-4/√2) = 2/√2
Vertical component of the resultant vector = (6/√2) + (4/√2) = 10/√2

Step 4: Find the magnitude of the resultant vector
Magnitude of the resultant vector = √[(2/√2)^2 + (10/√2)^2] = √52 = 2√13 km

Step 5: Find the angle of the resultant vector with the east
Tanθ = (10/√2)/(2/√2) = 5
θ = tan^-1(5) = 78.69^

Step 6: Conclusion
The distance of the point from the starting point is 2√13 km and the angle of the straight line joining its initial and final positions with the east is 78.69^
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A car travels 6 km towards the north at an angle 45^ to the east and then travels distance of 4 km towards the north at an angle 135^ to the east. How far is the point from the starting point? What angle does the straight line joining its initial and final positions makes with the east?
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