Introduction to Resultant Vectors

Hello everyone! Today, we’re going to learn about resultant vectors and how to graph them using the triangle method.

A vector is a quantity that has both magnitude (size) and direction. For example, if you walk 5 metres to the north, that’s a vector because it tells us how far and in which direction.

The Triangle Method

The triangle method is a simple way to find the resultant vector when you have two or more vectors.

Steps to Graph a Resultant Vector:

  1. Draw the First Vector: Start by drawing your first vector. Use a ruler and a protractor to make sure it’s accurate.
  2. Draw the Second Vector: From the tip (end) of the first vector, draw the second vector. The direction should be correct, and the length should match the magnitude.
  3. Complete the Triangle: Now, draw a line from the tail (start) of the first vector to the tip of the second vector. This line is your resultant vector.
  4. Label the Resultant Vector: Finally, label the resultant vector with an arrow to show its direction.

Example

Let’s say:

  • Vector A is 4 cm to the right (east).
  • Vector B is 3 cm upwards (north).
  1. Draw a 4 cm line to the right.
  2. From the end of that line, draw a 3 cm line upwards.
  3. Now, draw a straight line from where you started to the tip of the last line. This is your resultant vector.

Key Rules

  • Always use a ruler for accuracy.
  • Use a protractor to measure angles if they are not at 90 degrees.
  • Remember that the direction matters! Make sure to draw arrows on your vectors.

Tips and Tricks

  • Use different colours for different vectors to make your diagram clearer.
  • Check your work by measuring the resultant vector and comparing it with your calculations.
  • Practice with different vectors to get the hang of it!

Practice Questions

Easy Level Questions

  1. What is a vector?
  2. Draw a vector of 5 cm to the north.
  3. Draw a vector of 3 cm to the east.
  4. If vector A is 2 cm to the left, where does the tip end up?
  5. Draw a triangle using vectors of 2 cm up and 2 cm right.
  6. How do you label a resultant vector?
  7. What tools do you need to graph vectors?
  8. If vector B is 4 cm down, what direction is it?
  9. Draw a vector of 6 cm to the south.
  10. What is the first step in the triangle method?

Medium Level Questions

  1. Draw vectors A (3 cm right) and B (4 cm up). What is the length of the resultant vector?
  2. If vector A is 5 cm at an angle of 30 degrees to the east, draw it.
  3. Draw vectors A (5 cm north) and B (5 cm east). What is the direction of the resultant?
  4. How can you check the accuracy of your resultant vector?
  5. If vector C is 3 cm down, what direction is it?
  6. Draw a triangle with vectors of 4 cm up and 3 cm left.
  7. If vector D is 6 cm to the right and E is 2 cm down, what is the resultant?
  8. Explain why the order of drawing vectors matters.
  9. Draw a vector of 7 cm at 45 degrees. What shape does it form?
  10. Given vectors of 2 cm left and 2 cm down, what is the shape formed?

Hard Level Questions

  1. Calculate the resultant of vectors A (5 cm at 60°) and B (7 cm at 30°).
  2. Draw the triangle for vectors of 8 cm south and 6 cm east. What is the resultant?
  3. If vector F is 10 cm at 120 degrees, how do you graph it?
  4. Find the length of the resultant vector for vectors of 3 cm up and 4 cm right.
  5. Explain how to find the angle of the resultant vector.
  6. Given vectors of 5 cm right and 12 cm down, draw and calculate the resultant.
  7. If vector G is 8 cm at 210 degrees, what direction is it?
  8. Draw vectors of 9 cm and 12 cm at right angles. Find the resultant.
  9. Create a triangle using vectors of 5 cm forward and 12 cm to the left. What is the resultant?
  10. If you have vectors of 10 cm at 45 degrees and 10 cm at 135 degrees, how do you graph the resultant?

Answers and Explanations

Easy Level Answers

  1. A vector is a quantity with both magnitude and direction.
  2. (Draw a 5 cm line pointing north.)
  3. (Draw a 3 cm line pointing east.)
  4. The tip ends up 2 cm to the left of the start.
  5. (Draw a triangle with vectors going up and right.)
  6. You label it with an arrow.
  7. You need a ruler, protractor, and pencil.
  8. It goes downwards.
  9. (Draw a 6 cm line pointing south.)
  10. Draw the first vector.

Medium Level Answers

  1. The resultant length is 5 cm (using Pythagoras’ theorem).
  2. (Draw 5 cm at 30 degrees).
  3. The resultant is northeast.
  4. Measure the resultant with a ruler.
  5. It goes downwards.
  6. (Draw the triangle.)
  7. The resultant is √(6²+2²).
  8. It ensures the correct direction.
  9. It forms a diagonal line.
  10. It forms a right triangle.

Hard Level Answers

  1. Use the cosine rule to find the length of the resultant.
  2. (Draw the triangle and calculate using Pythagorean theorem.)
  3. Graph it from the origin using a protractor.
  4. Resultant length is 5 cm.
  5. Use trigonometry to find the angle.
  6. The resultant is √(5²+12²).
  7. It’s in the third quadrant.
  8. (Draw right-angle triangle; calculate using Pythagorean theorem.)
  9. Use triangle inequality to find the resultant.
  10. Use vector addition.

Feel free to ask questions if you need more help with vectors!