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In this lesson, we will learn how to use vector operations to solve real-world problems on different topics, such as navigation and physics.

Q1:

A truck moved 150 km due east and then 200 km at a direction of 6 0 ∘ . Determine the truck’s displacement, giving its magnitude to the nearest kilometre and its direction to the nearest minute.

Q2:

Two children are throwing a ball to each other straight across the back seat of a car, perpendicular to the direction of travel of the car. The ball is being thrown at 10 mph relative to the car, and the car is traveling in a straight line at 25 mph along the road. If one child misses the ball, letting it fly out of the window, find the angle it makes relative to the car’s forward direction ignoring wind resistance. Give your answer in degrees, correct to one decimal place if necessary.

Q3:

Two cars are moving on the same straight road with velocities of 87 km/h and 85 km/h, respectively. Given that ⃑ 𝑒 is a unit vector in the direction of movement of the first car, determine the relative velocity of the first car relative to the second, given that both cars are moving in the same direction.

Q4:

Two cars are moving on the same straight road with velocities of 17 km/h and 50 km/h, respectively. Given that ⃑ 𝑒 is a unit vector in the direction of movement of the first car, determine the relative velocity of the first car relative to the second, given that both cars are moving in the same direction.

Q5:

Velocity is a vector quantity which combines speed and direction. For example, the velocity of a section of the Mississippi river near New Orleans is 3 miles per hour east.

What is the magnitude of this velocity?

What is the direction of this velocity?

Q6:

A man starts walking from home and walks 3 miles at 2 0 ∘ north of west, then 5 miles at 1 0 ∘ west of south, then 4 miles at 1 5 ∘ north of east. If he walked in a straight line back home, how far would he have to walk and in what direction? Give your answers correct to three decimal places.

Q7:

Two children are throwing a ball to each other straight across the back seat of a car, perpendicular to the direction of travel of the car. The ball is being thrown at 8 mph relative to the car, and the car is traveling in a straight line at 45 mph along the road. If one child misses the ball, letting it fly out of the window, find the angle it makes relative to the car’s forward direction ignoring wind resistance. Give your answer in degrees, correct to three decimal places if necessary.

Q8:

Two cars 𝐴 and 𝐵 are moving in opposite directions on the same road at 62 km/h and 31 km/h, respectively. Given that e is a unit vector in the direction of movement of car 𝐴 , determine the velocity of car 𝐴 relative to car 𝐵 .

Q9:

Two cars 𝐴 and 𝐵 are moving in opposite directions on the same road at 13 km/h and 82 km/h, respectively. Given that e is a unit vector in the direction of movement of car 𝐴 , determine the velocity of car 𝐴 relative to car 𝐵 .

Q10:

Find the values of 𝑘 , 𝑚 , and 𝑛 that make the two vectors ⃑ 𝐴 = 𝑘 − 2 , 𝑚 − 2 , 4 2 and ⃑ 𝐵 = 3 , 2 , 𝑛 2 equal.

Q11:

A police car was moving along a horizontal highway at 47 km/h. It used a radar to measure the speed of a truck moving in the same direction. Given that the reading on the radar was 50 km/h, determine the actual speed of the truck.

Q12:

A police car was moving along a horizontal highway at 38 km/h. It used a radar to measure the speed of a truck moving in the opposite direction. Given that the reading on the radar was 70 km/h, determine the actual speed of the truck.

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