Draw a vector N to represent a wind blowing north at 55 km per hour for one hour and a vector M representing a car travelling west that has gone 45 km.

The question asks to "multiply" two 2D vectors. In vector algebra, the standard multiplication that results in a scalar is the dot product.

Let $\vec{v} = \begin{pmatrix} 5 \\ -4 \end{pmatrix}$ and $\vec{u} = \begin{pmatrix} -8 \\ 7 \end{pmatrix}$.

The dot product of two vectors $\vec{a} = \begin{pmatrix} a_x \\ a_y \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} b_x \\ b_y \end{pmatrix}$ is given by the formula $\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y$.

The result of multiplying vector $\vec{v}$ by vector $\vec{u}$ (dot product) is:
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• Vector N (wind): The wind blows north at 55 km per hour for one hour, so its magnitude is $55 \text{ km}$. Its direction is North.
• Vector M (car): The car travels west and has gone 45 km, so its magnitude is $45 \text{ km}$. Its direction is West.

The grid is 8x8 squares. A convenient scale that allows both vectors to be represented clearly within the grid is to let each square represent $10 \text{ km}$.

• For Vector N: $55 \text{ km} \div 10 \text{ km/square} = 5.5 \text{ squares}$.
• For Vector M: $45 \text{ km} \div 10 \text{ km/square} = 4.5 \text{ squares}$.

Assume the bottom-left corner of the grid is the origin $(0,0)$ and the top-right is $(8,8)$.

• Vector N (Wind):
  • Start at a point, for example, the center of the bottom edge of the grid, which can be represented as $(4,0)$.
  • Draw an arrow pointing vertically upwards (North) from this starting point.
  • The length of the arrow should be 5.5 squares. The arrow would end at $(4, 5.5)$.
• Vector M (Car):
  • Start at a different point, for example, the center of the right edge of the grid, which can be represented as $(8,4)$.
  • Draw an arrow pointing horizontally leftwards (West) from this starting point.
  • The length of the arrow should be 4.5 squares. The arrow would end at $(3.5, 4)$.