スカラーとベクトル

Which way?

• Walk $3\ \text{km}$ out and $3\ \text{km}$ back: you travel $6\ \text{km}$, but end up where you started.
• The distance 距离 is $6\ \text{km}$; the displacement 位移 is $0$.
• The difference is direction — and that is the whole idea here.

Scalars 标量 and vectors 矢量

• A scalar has size only.
• A vector has size and direction.

The SI prefixes climb in steps of a thousand, from pico to giga

Which is which?

• Scalars: mass, time, temperature, energy, work, power, distance, speed, density 密度.
• Vectors: displacement, velocity, acceleration, force (incl. weight), momentum.

A quick test

• Ask: "in which direction?"
• "In which direction is the temperature?" makes no sense → temperature is a scalar.
• "In which direction is the velocity?" makes sense → velocity is a vector.

Vectors are arrows

• Draw a vector as an arrow.
• Its length (to scale) shows the magnitude; its direction shows the direction.

A vector is an arrow — its scaled length is the magnitude and the way it points is the direction

Adding vectors: tip to tail

• Draw the arrows tip to tail.
• The resultant 合矢量 goes from the first tail to the last tip.

Subtracting vectors

• To find $\vec{X} - \vec{Y}$, add the reverse of $\vec{Y}$: $\vec{X} + (-\vec{Y})$.
• The reverse has the same size but points the opposite way.

Resolving into components 分量

• Any vector splits into two perpendicular 垂直 parts.
• For a vector $v$ at angle $\theta$ to the horizontal: $v_{\text{H}} = v\cos\theta$, $v_{\text{V}} = v\sin\theta$.

On a slope

• On a slope of angle $\theta$, split the weight along and across the slope.
• Along: $W_{\parallel} = W\sin\theta$. Across: $W_{\perp} = W\cos\theta$.

When do you resolve?

• Whenever you need how much of a vector acts in one direction.
• Example: the part of a force pulling along a ramp, or the up and across parts of a thrown ball's velocity.

You've got it