1.1 Scalars and Vectors

A scalar is a quantity described completely by one number and a unit. Distance, speed, mass, time and energy are scalars. Saying "the run took 12 s" needs no direction.

A vector has a magnitude (its size, never negative) and a direction. Position, displacement, velocity, acceleration, force and momentum are vectors. "20 m/s" is a speed, but "20 m/s north" is a velocity.

Several pairs look alike but aren't. Distance is the total length of the path you travel, a scalar. Displacement is the straight-line change in position from start to finish, a vector. Walk once around a 400 m track and your distance is 400 m, but your displacement is zero. Speed is how fast; velocity is how fast and which way.

In one dimension, a sign carries the direction. If you pick right as positive, a velocity of −3 m/s means 3 m/s to the left. A negative velocity is not "less" motion than a positive one.

You can describe a vector by its magnitude and an angle, like "5.0 m at 37° north of east". Or you can give its components: how far it reaches along x and along y.

Unit vector notation uses $\hat{i}$, $\hat{j}$ and $\hat{k}$, arrows of length 1 pointing along +x, +y and +z. The vector $\vec{A} = 4\hat{i} - 3\hat{j}$ means 4 units in the +x direction and 3 units in the −y direction.

To go from magnitude and angle to components, use trig. If θ is measured from the +x axis, $A_x = A\cos\theta$ and $A_y = A\sin\theta$. If the angle is measured from some other line, draw the triangle and decide which side is opposite the angle. Don't just memorize "x is cosine".

To go back, use the Pythagorean theorem and the inverse tangent:

$$A = \sqrt{A_x^2 + A_y^2}, \qquad \theta = \tan^{-1}\!\left(\frac{A_y}{A_x}\right)$$

Your calculator's inverse tangent only returns angles between −90° and 90°. Sketch the vector so you know which quadrant it's in, and adjust the angle if it points left.

Graphically, you add vectors tip to tail: draw the second vector starting where the first one ends. The resultant (the sum) runs from the start of the first to the tip of the last. Order doesn't matter.

With components, adding is just adding: x-parts with x-parts and y-parts with y-parts. If $\vec{A} = A_x\hat{i} + A_y\hat{j}$ and $\vec{B} = B_x\hat{i} + B_y\hat{j}$, then $\vec{A} + \vec{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$.

Subtracting a vector means adding its opposite: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$, where $-\vec{B}$ has the same size as $\vec{B}$ but points the other way. You'll subtract vectors constantly, because a change like $\Delta\vec{v} = \vec{v}_f - \vec{v}_i$ is a subtraction.

Multiplying a vector by a positive scalar stretches or shrinks it without changing its direction. Multiplying by a negative scalar also flips it. That's why $\vec{p} = m\vec{v}$ points the same way as $\vec{v}$.

Expect to break vectors into components, add them, and find a resultant's magnitude and direction. Numerical work stays in one or two dimensions. Three-dimensional vectors (with $\hat{k}$) show up mostly in descriptions and, later, in the cross product for torque.

Two ways of multiplying vectors come later: the dot product, which gives a scalar and is used for work (3.2) and power (3.5), and the cross product, which gives a vector and is used for torque and angular momentum.