1.4 - Adding and subtracting vectors

Scalars, vectors and direction

Can you walk $100\,\mathrm{m}$ and finish with zero displacement? Walk once around a closed track and you have done exactly that. The distance records the journey's length; the displacement records the change in position.

A scalar has magnitude without an associated direction. Distance, speed, mass, time, energy and power are scalar quantities. A vector has both magnitude and direction. Displacement, velocity, acceleration, force and momentum are vectors. You will study each in its physical setting as the course develops.

Giving a speed of $6.0\,\mathrm{m\,s^{-1}}$ says how fast something moves. Giving a velocity of $6.0\,\mathrm{m\,s^{-1}}$ due north also says which way. Two objects can have equal speeds and different velocities. Direction is part of the physical information, rather than an optional label.

Represent a vector by an arrow. Its length, on a chosen scale, represents magnitude; its arrowhead represents direction. In equations, bold symbols such as $\mathbf A$ distinguish a vector from its magnitude $A$. Equal vectors have equal magnitudes and directions. For the addition of displacement vectors, you may move the arrows on the page without changing their lengths or directions.

A negative sign reverses a vector: $-\mathbf A$ has the same magnitude as $\mathbf A$ but the opposite direction. It does not make the magnitude negative. Scalars can also have signed values in some contexts; a negative Celsius temperature does not turn temperature into a vector.

The distinction matters before any calculation.

Adding coplanar vectors

A resultant is the single vector with the same combined effect as the vectors being added. For two successive displacements, it points directly from the initial position to the final position, regardless of the route between them.

To add $\mathbf A$ and $\mathbf B$, put the tail of $\mathbf B$ at the head of $\mathbf A$. Draw the resultant $\mathbf R=\mathbf A+\mathbf B$ from the first tail to the final head. This is the tip-to-tail method. The vectors must represent the same kind of quantity: adding a force to a velocity has no meaning.

For a scale drawing, choose a scale such as 1 cm for 1 m, preserve the angles and measure the resultant. Accurate calculation is preferable when sufficient numerical information is supplied. The parallelogram method gives the same result: put both tails together and draw their resultant along the diagonal from that common tail. In the force examples here, the forces act on the same object and we calculate their combined translational effect. Turning effects need the points where the forces act as well; you will study those separately.

[DIAGRAM: asset_name: Tip-to-tail addition; asset_slug: coplanar-vector-addition; caption: The resultant joins the starting point to the final tip; description: Two coordinate diagrams show a blue vector followed by a sage vector, with a clay resultant from first tail to final head. The left uses 3 metres east then 4 metres north, giving 5 metres at 53 degrees north of east. The right uses 8 newtons east then 6 newtons at 60 degrees north of east, showing the 120-degree interior triangle angle and a resultant of 12.2 newtons at 25.3 degrees north of east.]

When the vectors are not perpendicular, Pythagoras alone is insufficient. Use the cosine and sine rules for a general triangle.

A drawing prevents you confusing the angle between directions with the interior angle.

For parallel vectors in the same direction, add magnitudes. For opposite directions, subtract magnitudes and keep the direction of the larger vector. These are limiting cases of the same vector rule. As the angle between vectors changes from $0^\circ$ to $180^\circ$, the resultant changes from $A+B$ to $|A-B|$.

Subtracting vectors and finding a change

A vector change compares a final vector with an initial vector. To subtract the initial vector, reverse it and add it: $\Delta\mathbf v=\mathbf v_{\mathrm f}-\mathbf v_{\mathrm i}=\mathbf v_{\mathrm f}+(-\mathbf v_{\mathrm i})$. Here the two velocities and their difference all have unit $\mathrm{m\,s^{-1}}$. This vector-change rule is not on the data sheet: learn it.

If both original vectors start at the same point on a diagram, their difference runs from the tip of the initial vector to the tip of the final vector. That arrow tells you what must be added to the initial vector to obtain the final one.

The most revealing case is a turn at unchanged speed.

[DIAGRAM: asset_name: The arrow for a velocity change; asset_slug: velocity-vector-subtraction; caption: Final minus initial points from the initial tip to the final tip; description: Initial velocity is a blue eastward arrow of 10 metres per second and final velocity a sage northward arrow of the same length, both with a common tail. A clay arrow joins the initial tip to the final tip, pointing north-west and labelled change in velocity, magnitude 14.1 metres per second. A small equation states initial velocity plus change equals final velocity.]

This way of thinking will be essential for acceleration and circular motion. A scalar difference of speeds cannot replace a vector difference of velocities.

Use a fresh turn to test the direction.

Quick Recap

• Scalars have magnitude; vectors also carry direction and follow vector addition.
• Add coplanar vectors tip to tail and draw the resultant from the first tail to the final head.
• Use suitable triangle geometry, including sine and cosine rules, to determine magnitude and direction.
• Subtract a vector by reversing it; final minus initial points from the initial tip to the final tip.

You can walk $100\,\mathrm{m}$ with zero displacement because a closed route ends at its starting position, even though the distance travelled is not zero.