A scalar is quantity, such as mass, speed, or temperature, which has a magnitude only. Any scalar can be represented by a single number, which may or may not be signed.
We represent a scalar quantity in algebra by a normal letter without extra ornamentation, e.g. m, x, v, t.
A vector is quantity, such as displacement, velocity, or force, which has both a magnitude and a direction. It takes as many numbers to represent a vector as the space has dimensions, 2 numbers in 2-D, 3 in 3-D, and so on. The most common representation is the Cartesian components of the line (x, y, z) but we can also use magnitude and angle versions such as (r, θ) in 2-D or (r, Θ, φ) in 3-D.
We represent a vector in diagrams by an arrow, a directed line segment. The length and orientation are significant but the location of the vector is usually not.
We represent a vector quantity in algebra by either a bold letter or by a letter with an underline or over-arrow, e.g. x, x, v, v, F, and F.
We add vectors graphically by placing them tip to tail and then constructing a segment from the free tail to the free tip.
We subtract vectors graphically by placing the tails together and then constructing the segment from free tip to free tip. The direction of the segment depends on the order of the vectors. If we are constructing a-b then the arrowhead goes on the end nearest the tip of a, the positive quantity in the formula.
We add vectors in Cartesian coordinate form by adding their Cartesian components. e.g. V1+V2=(V1x+V2x, V1y + V2y).
A particle moving in 2 or more dimensions executes its motions in orthogonal directions completely independently. In 2-D for example, the vertical and horizontal motions are completely independent. Each Cartesian component evolves under the influence of its acceleration independently of the others.
In the common case of a particle moving under the influence of the constant gravitational acceleration near the surface of the earth we have
| x(t) = x0 + v0x×t | y(t) = y0 + v0y×t - ½g×t2 |
| vx(t) = v0x | v(t) = v 0 + a×t |
Under these circumstances the motion is symmetrical (unless outside objects intervene). The speed is the same at the same height whether going up or coming down and each segment of the trip up takes the same time as the corresponding segment of the trip down.
The range of particle launched on level ground with speed v0 at an angle θ to the horizontal is
Range = v02sin(2θ)/g