Vectors: dot and cross product: watch it solved step by step

Dot product, cross product, lengths, the angle between two vectors, projection and parallelogram area. Level: A-level. Type any problem of this kind and the Lab works it out and animates every move, with a plain-words explanation of why.

Worked example: (3, 4) and (-4, 3)

1. The vectors. A vector has a size and a direction. Written in components it says how far to go along each axis.
2. Dot product. Multiply matching components and add. The answer is a number (a scalar), and it is 0 exactly when the vectors are perpendicular.
3. Magnitudes (lengths). The length of a vector is the square root of the sum of the squares of its components (Pythagoras).
4. The angle between them. The dot product also equals |a||b|, so rearrange for the angle.
5. The 2D cross product (a number). In 2D, a_1b_2 - a_2b_1 is the signed area of the parallelogram the vectors span.
6. Projection of a onto b. How much of a points along b: the shadow of a on b.

Answer: a·b = 0; |a| = 5, |b| = 5; angle = 90°; parallelogram area = 25

Practise vectors: dot and cross product with problems the Lab marks for you.