Solve Problems with Vectors

Lesson 3 of 12: VM Cluster

In this lesson:

• Represent velocity, force, and displacement as vectors
• Find the resultant of two simultaneous velocity vectors
• Distinguish displacement from total distance

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Represent velocity, force, and displacement as directed segments or component vectors
2. Set up a vector diagram for velocity-combination problems
3. Compute the resultant vector magnitude using the Pythagorean theorem
4. Distinguish displacement (vector) from total distance (scalar)

Recall: Component Form from Lesson 2

• Component vector: — horizontal and vertical displacement
• Positive : rightward; negative : leftward
• Positive : upward; negative : downward
• Magnitude:

Can you read and describe the direction it points?

What Is the Ferry's Actual Velocity?

A ferry heads due east at 5 m/s.
A river current flows due south at 3 m/s.

Does the ferry actually travel due east — or somewhere else?

Think about this before you start drawing.

Drawing the Ferry and Current Vectors

• Ferry velocity: m/s — 5 m/s east, no vertical movement
• Current velocity: m/s — no horizontal movement, 3 m/s south

Reading Velocity into Component Form

Any velocity can be expressed as where:

• = eastward speed (negative = westward)
• = northward speed (negative = southward)

Example: "400 mph at 45° northeast" → mph, mph.

Ferry and Current: Labeling Each Vector

Ferry: m/s — 5 m/s east, no vertical component

Current: m/s — 3 m/s south, no horizontal component

Verify: m/s ✓; m/s ✓

Check-In: Express Velocity as Components

A plane flies 400 mph at 45° northeast.

Express the plane's velocity in component form .

Use .

Placing Vectors Head to Tail for Resultant

What Happens If the Current Grows Stronger?

The current strengthens from 3 m/s to 6 m/s: m/s.

Predict: Does the resultant speed increase, decrease, or stay the same?

Commit to an answer before the next slide.

Resultant Magnitude from Perpendicular Vectors

When two velocity vectors are perpendicular, the resultant forms a right triangle:

This is the same magnitude formula from Lesson 2 — applied to velocities.

Resultant Speed Is Not the Sum of Speeds

Wrong approach: m/s ✗

Correct approach: m/s ✓

The sum of the legs (8) is always longer than the hypotenuse ().

Ferry Resultant: Full Worked Example

Given: ferry m/s; current m/s

Resultant speed:

Direction: south of east

Worked Example: Boat Crossing a River

Boat: 8 m/s due north. Current: 6 m/s due east.

Note: — the resultant is less than the scalar sum.

Check-In: Find the Resultant Speed

A swimmer crosses a river at 2 m/s perpendicular to the banks.
The current runs parallel to the banks at 1.5 m/s.

Find the swimmer's actual speed.

Draw the right triangle before applying the formula.

Force and Displacement Are Vectors Too

Beyond velocity, two more physical quantities are vectors:

• Force — has magnitude (Newtons) and direction (angle of application)
• Displacement — straight-line vector from start to finish; distinct from total distance traveled

Both use component form and the magnitude formula.

Force Components at an Angle

• N — horizontal push
• N — vertical lift

Displacement and Distance Are Not the Same

• Distance: 4 + 3 = 7 km (how far you walked)
• Displacement: km (how far from start)

Finding Displacement in a Three-Leg Journey

Journey: 3 km east → 4 km north → 2 km west

• Total distance: km (scalar)
• Net displacement: km (net x = ; net y = )
• Magnitude: km

Find and Fix the Resultant Error

A student computed the following:

"Boat goes 8 m/s north; current is 6 m/s east. Resultant speed = 8 + 6 = 14 m/s."

What did this student do wrong?

Write the correct resultant speed and explain.

End-to-End Practice: Plane in a Crosswind

A plane flies 300 mph due north.
A crosswind blows 40 mph due west.

1. Draw the vector diagram
2. Find the resultant speed (to the nearest mph)
3. Estimate the direction angle

Key Takeaways: Vectors in the Real World

✓ Velocity, force, displacement are all vector quantities — magnitude + direction

✓ Resultant: place vectors tail-to-tip; resultant runs from first tail to last tip

✓ Perpendicular resultant: — not

✓ Displacement ≠ distance: displacement is the straight-line start-to-end vector

Next Lesson: Adding Vectors Algebraically

You combined velocity vectors geometrically (tail-to-tip) and computed resultants.

Next (VM.B.4): Add vectors algebraically — component by component:

The geometry you built today is the foundation for the algebra ahead.