Scalars vs Vectors

Let's say I started at my house, and I walked two miles. Where am I?

I could be two miles east of my house, or two miles west of my house, or maybe even back at my house — I could have walked one mile east and then one mile back.

But if I tell you I walked two miles east, then (assuming you know where I started), you know exactly where I am.

"Two miles" is a scalar — it tells you only the magnitude (how much). "Two miles east" is a vector — it tells you both magnitude and direction.

Key difference

A scalar has magnitude only (just "how much"). A vector has magnitude and direction ("how much" and "which way"). Vectors are written with arrows, like $\vec{v}$. The size (magnitude) of a vector is written $|\vec{v}|$ and is nonnegative.

Common scalars

• Distance ($d$): example $d = 200$ m — total path length
• Speed ($v$): example $v = 15$ m/s — how fast
• Mass ($m$): example $m = 2.0$ kg — amount of matter
• Time ($t$): example $t = 5.0$ s — duration
• Temperature ($T$): example $T = 300$ K — hot/cold level

Common vectors

• Displacement ($\vec{d}$): example $\vec{d} = 50$ m east — straight-line change in position
• Velocity ($\vec{v}$): example $\vec{v} = 15$ m/s east — how fast and which way

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