Vectors cannot be added like ordinary numbers unless they act along the same line
Scalars & Vectors - OCR A-Level Physics
Key Definition
Triangle of vectors
A method for adding two vectors: draw the first vector to scale, then draw the second vector starting at the nose of the first (nose-to-tail). The resultantThe single vector that has the same effect as two or more vectors acting together. is the third side of the triangle, drawn from the tail of the first vector to the nose of the second.
A method for adding two vectors: draw the first vector to scale, then draw the second vector starting at the nose of the first (nose-to-tail). The resultantThe single vector that has the same effect as two or more vectors acting together. is the third side of the triangle, drawn from the tail of the first vector to the nose of the second.
Diagram pending
Triangle of vectors. Vector $\mathbf{A}$ drawn from origin pointing right; vector $\mathbf{B}$ drawn from the nose of $\mathbf{A}$ pointing up-right. Dashed resultant $\mathbf{R}$ drawn from origin to the nose of $\mathbf{B}$, with arrowhead. Angle between $\mathbf{R}$ and the horizontal labelled.
Will be replaced with a GeoGebra SVG in stream 2.
- Vectors cannot be added like ordinary numbers unless they act along the same line. Two forces of 3 N and 4 N at right angles give a resultant of 5 N, not 7 N.
- Step 1: choose a scale (e.g. $1 \text{ cm} = 5 \text{ N}$) and draw the first vector to that length, in the correct direction.
- Step 2: from the nose (arrowhead) of the first vector, draw the second vector to the same scale, in its own direction.
- Step 3: the resultant is the arrow from the tail of the first vector to the nose of the second. Measure its length and convert back using the scale; measure its angle with a protractor.
- For calculation (not scale drawing), use the cosine ruleFor a triangle with sides $a$, $b$, $c$ and angle $C$ opposite $c$: $c^2 = a^2 + b^2 - 2ab\cos C$. for the magnitude and the sine ruleFor a triangle: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. for the direction.
$$R^2 = A^2 + B^2 - 2AB\cos\phi$$
where $\phi$ is the interior angle of the triangle (the angle between the two vectors when drawn nose-to-tail, which equals $180^\circ$ minus the angle between them when drawn from the same point).
Examiner Tips and Tricks
- In a scale drawing question, always state your scale on the page (e.g. "$1 \text{ cm}$ represents $5 \text{ N}$"). Use a sharp pencil and a protractor.
- Measure the resultant length, then convert back using your scale to give the answer in the correct units.
- Examiners accept any answer within $\pm 5\%$ of the calculated value, provided your scale is stated.
- If the question allows either scale drawing or calculation, calculation is usually quicker and more accurate.
Common Mistake
MEDIUM
Wrong: Drawing both vectors from the same point when using the triangle method, then drawing the resultant between their noses. Or simply adding magnitudes: $3 \text{ N} + 4 \text{ N} = 7 \text{ N}$ for vectors at an angle.
Right: Triangle method: vectors are drawn nose-to-tail. The second vector starts at the nose of the first. Drawing both from the same origin is the parallelogram method, not the triangle method. Magnitudes are only added arithmetically when both vectors point the same way.
Right: Triangle method: vectors are drawn nose-to-tail. The second vector starts at the nose of the first. Drawing both from the same origin is the parallelogram method, not the triangle method. Magnitudes are only added arithmetically when both vectors point the same way.