What is Vector?

A scalar, such as mass or time, has size only. A vector is added by the triangle rule: place the tail of the second at the head of the first. In components, $\vec{a} = a_x\hat{i} + a_y\hat{j}$, and its magnitude is $\sqrt{a_x^2 + a_y^2}$. The dot product $\vec{a}\cdot\vec{b} = ab\cos\theta$ gives a scalar. The cross product gives a vector at right angles to both.