Motion in a PlaneScalars and Vectors
Physical quantities come in two kinds. A scalar has only a magnitude: one number with its unit says everything there is to say. The distance between two towns, the mass of a cricket ball, the temperature of your classroom and the time on a clock are all scalars. Scalars combine by the ordinary rules of arithmetic.
Riya puts a ribbon border round a rectangular rangoli board 1.0 m long and 0.5 m wide. She needs 1.0 m + 0.5 m + 1.0 m + 0.5 m =
On a summer day in Nagpur the maximum temperature is 35.6 °C and the minimum is 24.2 °C, so the day's range is
Tip
Add or subtract scalars only when they have the same unit: metres to metres, °C to °C. Multiplying and dividing scalars with different units is fine — that is how mass ÷ volume becomes a density.
Size and Direction
A vector quantity has both a magnitude and a direction, and it combines with other vectors by the triangle law (or, equivalently, the parallelogram law) of addition, which you will meet in Section 3.4. Displacement, velocity, acceleration and force are vectors.
In print a vector is written in bold type, v; by hand, with an arrow over the letter, . Its magnitude, also called its absolute value, is written |v| or simply v in ordinary type. So |A| = A, |p| = p, and so on.
Drag each card into its bucket.
Speed and velocity are the classic pair: "40 km/h" is a speed, a scalar, while "40 km/h towards Pune" is a velocity, a vector. In the same way, distance is a scalar and displacement is a vector.
Position and Displacement Vectors
To say where something is in a plane, first choose a reference point O, the origin. If an object is at P at time t, the arrow from O to P is its position vector at that time, r = OP. The length of the arrow is how far P is from O, and its direction is the direction in which P lies as seen from O.
If the object moves from P to a new position P′, the arrow from P to P′ is its displacement vector. It joins the start to the finish in a straight line, whatever route the object actually took between them.
Aman's town has a grid of lanes 100 m apart. He lives at P and his school is at Q. Tap crossings to walk him around the town, and complete the three missions.
Three very different walks. The position vector r of P stayed put, and the displacement only ever cared about where Aman ended up: from P to Q it is 500 m long, whether he walked 500 m or 1500 m along the grey route. So the magnitude of the displacement is less than or equal to the path length; the two are equal only when the object moves along a straight line without turning back.
Equality of Vectors
Two vectors A and B are equal if, and only if, they have the same magnitude and the same direction. Where they are drawn does not matter: in this chapter vectors are free vectors, so sliding one parallel to itself leaves it unchanged.
That gives a test, the one in Fig. 3.2: slide B parallel to itself until its tail sits on the tail of A. If the tips then land on each other too, the vectors are equal.
Drag the green vectors by their tails and test both against the blue vector A.
The vector B passed the test, so B = A. The vector B′ has exactly the same length, √13 units, yet its tip missed: B′ ≠ A, because its direction is different.
Note
For some vectors the location does matter. The same push on a door does very different things at the hinge and at the handle. A vector whose line of action matters is called a localised vector. Until the chapter on rotation, our vectors are free.
Speed Round: Scalar or Vector?
Ask of each quantity: does "in which direction?" make sense, and would two of them add like arrows? Get five right to move on.