1. Determine the resultant of each vector sum
a) 45 km/h vertically and then 75 km/h horizontally.
b) 300 m east and then 400 m N45¡ÆE
45 km/h vertically and then 75 km/h horizontally
--->45(cos90°, sin90°) + 75(cos0, sin0)
= (45(0,1) + 75(1,0)
= (0,45) + (75,0)
= (75,45)
magnitude = √(75^+45^2) = √7650
= 15√34
direction vector: tanØ = 45/75
Ø = appr 30.96°
or
you have a right-angled triangle, and the hypotenuse is √(75^2+45^2)
b) resultant = 300(-1,0) + 400(cos45,sin45)
= (300, 0) + (200√2 , 200√2)
= ( 200√2 + 300, 200√2)
follow the same steps I used for the first part
a) To determine the resultant of the vector sum, we need to add the vertical and horizontal components separately.
For the vertical component, the speed is 45 km/h vertically. Since there is no horizontal component, the vertical component is the full vector.
For the horizontal component, the speed is 75 km/h horizontally. Since there is no vertical component, the horizontal component is the full vector.
So, the resultant vector is the vector sum of 45 km/h vertically and 75 km/h horizontally.
b) To determine the resultant of the vector sum, we need to find the components of each vector and add them together.
For the first vector, 300 m east, the horizontal component is 300 m and the vertical component is 0 m.
For the second vector, 400 m N45°E, we need to break it down into its horizontal and vertical components. To do this, we use trigonometry.
The horizontal component is given by 400 m multiplied by the cosine of 45°. So, the horizontal component is 400 m * cos(45°) = 400 m * 0.7071 = 282.84 m.
The vertical component is given by 400 m multiplied by the sine of 45°. So, the vertical component is 400 m * sin(45°) = 400 m * 0.7071 = 282.84 m.
Now, we can add the horizontal and vertical components together to find the resultant vector.
Horizontal component: 300 m + 282.84 m = 582.84 m.
Vertical component: 0 m + 282.84 m = 282.84 m.
So, the resultant vector is 582.84 m east and 282.84 m north.
To determine the resultant of each vector sum, we need to add the vectors using vector addition. Vector addition involves finding the components of each vector and then adding them to find the resulting vector.
a) 45 km/h vertically and then 75 km/h horizontally:
To add these vectors, we need to find their horizontal and vertical components. The 45 km/h vector has no horizontal component (since it is vertical), and the 75 km/h vector has no vertical component (since it is horizontal).
For the vertical component:
The 45 km/h vector is vertical, so its vertical component is 45 km/h.
For the horizontal component:
The 75 km/h vector is horizontal, so its horizontal component is 75 km/h.
Now, we can add the components to find the resultant vector:
Vertical component = 45 km/h
Horizontal component = 75 km/h
Using Pythagoras theorem, we can find the magnitude of the resultant vector:
Resultant magnitude = sqrt((vertical component)^2 + (horizontal component)^2)
Resultant magnitude = sqrt((45 km/h)^2 + (75 km/h)^2)
To determine the direction of the resultant vector, we use trigonometry. We can find the angle using tangent:
Angle = arctan(vertical component / horizontal component)
Angle = arctan(45 km/h / 75 km/h)
b) 300 m east and then 400 m N45°E:
To add these vectors, we need to determine their components relative to a common reference direction (usually the positive x and y axes).
For the 300 m east vector:
Its x-component is 300 m, and its y-component is 0 since it is purely in the east direction.
For the 400 m N45°E vector:
We need to resolve it into its x and y components. The x-component will be the component in the east direction, while the y-component will be the component in the north direction. We can use trigonometry to find these components.
The angle between the vector and the east direction is 45°, so the x-component can be found using cosine:
x-component = 400 m * cos(45°)
The angle between the vector and the north direction is also 45°, so the y-component can be found using sine:
y-component = 400 m * sin(45°)
Now, we can add the components to find the resultant vector:
x-component = 300 m + x-component from N45°E vector
y-component = y-component from N45°E vector
Using Pythagoras theorem, we can find the magnitude of the resultant vector:
Resultant magnitude = sqrt((x-component)^2 + (y-component)^2)
Resultant magnitude = sqrt((300 m + x-component from N45°E vector)^2 + (y-component from N45°E vector)^2)
To determine the direction of the resultant vector, we use trigonometry. We can find the angle using tangent:
Angle = arctan(y-component / x-component)
Angle = arctan((y-component from N45°E vector) / x-component)