Vectors – sum, difference, scalar multiple and dot product

What it shows

Vectors in the plane can be combined both geometrically and with coordinates. The sum u + v is the diagonal of the parallelogram built on u and v and equals (x₁ + x₂; y₁ + y₂). The difference u − v runs from the tip of v to the tip of u. A scalar multiple k·u is parallel to u, |k| times as long, and reverses direction when k is negative. The dot product u·v = x₁x₂ + y₁y₂ = |u||v|cos(u, v) is a number, not a vector: positive for an acute angle, zero for perpendicular vectors and negative for an obtuse angle.

How to use

Pick the Starting operation, then drag the tips of u and v on the grid, or set their coordinates and the Number k. For the sum and the difference, compare the drawing with the coordinate calculation in the information line. For k·u, try positive, negative and fractional values of k. For the dot product, move v around u and watch the sign of u·v and the green projection as the angle passes 90°.

Parameters you can change

• u: x-coordinate x₁ -6–6
• u: y-coordinate y₁ -6–6
• v: x-coordinate x₂ -6–6
• v: y-coordinate y₂ -6–6
• Number k (for k·u) -3–3
• Starting operation Sum u + v, Difference u − v, Scalar multiple k·u, Dot product u·v

Questions to explore

1. With u = (4; 1), find a vector v such that u·v = 0. How many such vectors fit on the grid?
2. How are u − v and v − u related, both in the drawing and in coordinates?
3. For which values of k does k·u point opposite to u, and what happens when k = 0?