Scalar Product
From the previous worksheet, we saw that , where is the angle between the two vectors. Before you discover some properties, answer the following questions.
Question 1: How did we define the scalar product, using the length of the projection of onto ?
The applet below shows two vectors, which are placed at the same initial point (since we can freely move vectors). The magnitudes of each vector is also shown, along with the angle between them, and the scalar product between them.
Answer the questions, and move around the points to check various cases, until you are convinced of the property.
Question 1: Write out the properties of the scalar product, in relationship to the angle between the two vectors. Note: There should be at least four properties.
Question 2: What is equal to?
Question 3: Justify your answers to questions 1 and 2 using the projections discussed previously.
Question 4: Prove or justify your answers to questions 1 and 2, using the definition of the scalar product.