Vectors

The video lesson for this topic is coming soon.

Essential Question

How do we combine quantities that have a direction?

Many quantities in physics, like force, velocity, and displacement, have a direction as well as a size, and direction changes how they add together. By the end of this lesson, you should be able to add vectors tip to tail and break a vector into horizontal and vertical components.

Vector or Scalar?

• Scalar: size only
• Vector: size and direction
• Force, velocity, displacement are vectors

A scalar has only a size, such as a mass of 5.0 kg or a temperature of 25 °C. A vector has a size, called its magnitude, and a direction, such as a displacement of 5.0 m east or a force of 10 N down. Speed is a scalar, but velocity, which includes direction, is a vector. Force, velocity, and displacement are the vectors you will use most in physics.

Drawing Vectors as Arrows

• Arrow length shows the size
• Arrow points in the direction
• Use a scale, like 1 square = 10 N

A vector is drawn as an arrow. The arrow points in the vector’s direction, and its length shows the magnitude using a scale. With a scale of 1 square = 10 N, a 40 N force east is an arrow 4 squares long pointing right, and a 20 N force north is 2 squares long pointing up. The start of the arrow is the tail and the pointed end is the tip.

Adding Vectors Tip to Tail

• Put the tail of B at the tip of A
• Resultant: start of A to end of B
• Order does not matter

To add vectors, place them tip to tail: draw the first vector, then move the second one, without changing its length or direction, so its tail starts at the first vector’s tip. The resultant, R, is the arrow from the tail of the first vector to the tip of the last. It shows the overall effect, such as where you end up after two moves. Adding the vectors in a different order gives the same resultant.

Same or Opposite Directions

• Same direction: add the sizes
• Opposite directions: subtract, keep the bigger direction

When two vectors point in the same direction, add their sizes: 3 m east + 4 m east = 7 m east. When they point in opposite directions, subtract the smaller size from the larger, and the resultant points in the direction of the larger vector: 6 m east + 2 m west = 4 m east. This is how you find the net force when two forces pull against each other.

Perpendicular Vectors

• 90° apart: a right triangle
• Resultant is the hypotenuse
• Use the Pythagorean theorem

When two vectors are perpendicular (90° apart), placing them tip to tail makes a right triangle, and the resultant is the hypotenuse. Use the Pythagorean theorem to find its size: R = √(A² + B²). For 3.0 m east and 4.0 m north, R = √(9.0 + 16) = 5.0 m, not 7.0 m. The direction of the resultant can be found with inverse tangent: tan⁻¹(4.0 ÷ 3.0) ≈ 53° north of east.

Breaking a Vector into Parts

• Horizontal part: R cos θ
• Vertical part: R sin θ
• θ measured from the horizontal

Any vector can be split into two perpendicular components: a horizontal part R_x and a vertical part R_y. The vector is the hypotenuse of the right triangle they form, so when the angle θ is measured from the horizontal, R_x = R cos θ and R_y = R sin θ. For example, 10 m at 30° has R_x = 10 cos 30° = 8.7 m and R_y = 10 sin 30° = 5.0 m. If the angle is measured from the vertical instead, sine and cosine switch places.

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Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Tug of War

In a tug of war, one team pulls the rope west with 380 N. The other team pulls east with 425 N. What is the net force on the rope?

1. Check directions: 380 N west and 425 N east point in opposite directions
2. Subtract: 425 N − 380 N = 45 N
3. Direction: the larger force points east
4. Answer: 45 N east

Example 2: Walk to School

A student walks 1.5 km east, then 0.80 km north. What is the size of the student’s displacement from home?

1. Check directions: east and north are perpendicular, so use the Pythagorean theorem
2. Set up: R = √((1.5 km)² + (0.80 km)²)
3. Square and add: 2.25 km² + 0.64 km² = 2.89 km²
4. Square root: R = √(2.89 km²) = 1.7 km
5. Check: longer than 1.5 km, shorter than 1.5 km + 0.80 km = 2.3 km
6. Answer: 1.7 km (about 28° north of east)

Example 3: Kicked Soccer Ball

A soccer ball is kicked at 22 m/s at an angle of 40° above the ground. Find the horizontal and vertical components of its velocity.

1. Identify: v = 22 m/s at θ = 40° above the horizontal
2. Horizontal component: v_x = (22 m/s)(cos 40°) = 16.85 m/s
3. Vertical component: v_y = (22 m/s)(sin 40°) = 14.14 m/s
4. Check: both are less than 22 m/s, and v_x > v_y because 40° is less than 45°
5. Answer: v_x = 17 m/s, v_y = 14 m/s

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Big Ideas

• Vectors have size and direction
• Add vectors tip to tail
• Perpendicular: Pythagorean theorem
• Components: R cos θ and R sin θ

Back to the essential question: quantities with direction are combined as vectors, by placing them tip to tail and drawing the resultant from start to finish. Same-direction vectors add, opposite ones subtract, and perpendicular ones combine with the Pythagorean theorem, while components (R cos θ and R sin θ) let you split a vector back into horizontal and vertical parts.