How to sketch the triangle of forces for two vectors and a resultant

Key concept: In vector questions, the diagram matters as much as the numbers. If two forces act at the same point, the safest sketch is usually a head-to-tail triangle or a parallelogram, depending on what the question asks for.

Step by step

To show the relationship between $F_1$, $F_2$, and the resultant $R$:

Step 1: Draw $F_1$

• Sketch $F_1$ as a vector at 36° above the horizontal.
• Its magnitude is 215 N.

Step 2: Draw $F_2$

• Sketch $F_2$ as a vector at 23° below the horizontal.
• Its magnitude is 173 N.

Step 3: Show the resultant

• The resultant $R$ is the single vector that gives the same effect as $F_1$ and $F_2$ together.
• Use the head-to-tail method: place the tail of $F_2$ at the head of $F_1$, or vice versa.
• Then draw $R$ from the tail of the first vector to the head of the second.

So the required triangle should have:

• one side for $F_1$
• one side for $F_2$
• the third side as $R$

The triangle is not to scale unless the question says so, so the key thing is the direction and the head-to-tail arrangement, not exact lengths.

Pitfall alert

A common mistake is to draw both forces from the same starting point and call that the triangle. That gives a parallelogram idea, which may be fine in some contexts, but for a triangle of forces you need the vectors placed head-to-tail. Also, keep the angle reference straight: elevation is above the horizontal, depression is below it.

Try different conditions

If the forces were acting in opposite directions along the same line, the diagram would be a straight line rather than a triangle. If the question later asks for the magnitude of $R$, you would normally resolve each force into horizontal and vertical components and then use Pythagoras or trigonometry.

Further reading

resultant force, head-to-tail method, vector triangle