How Do You Calculate Dot Products in Geometric Problems? (2024)

In summary, the conversation is about the dot product in Euclidean space, particularly in the context of a given problem with p.q+p.r. The attempt at a solution involves expanding p.(q+r) and using the fact that ABE is an equilateral triangle to determine the magnitude of q. However, the answer scheme states that p.q=4.5, not 9 as the attempt at a solution suggests. The expert suggests starting with the geometric definition of the dot product to deduce the correct values of p.q and p.r.

1. What is the meaning of expanding p.(q+r)?

Expanding p.(q+r) is a mathematical expression that involves the use of parentheses to group numbers or variables and perform operations on them. In this case, p, q, and r are variables that represent numbers or other mathematical expressions.

2. How do you expand p.(q+r)?

To expand p.(q+r), you need to use the distributive property, which states that a(b+c) = ab + ac. In this case, p is being multiplied by the sum of q and r, so we can expand it as p.q + p.r.

3. What is the purpose of expanding p.(q+r)?

Expanding p.(q+r) helps us simplify mathematical expressions by breaking them down into smaller, more manageable parts. This can make it easier to solve equations or manipulate expressions to find their solutions.

4. Can you give an example of expanding p.(q+r)?

Sure, let's say we have the expression p.(2+3). Using the distributive property, we can expand it as p.2 + p.3, which simplifies to 2p + 3p. Similarly, if we have p.(x+y), we can expand it as p.x + p.y.

5. Are there any other ways to expand p.(q+r) besides using the distributive property?

Yes, there are other methods such as the FOIL method, which is commonly used to expand expressions with two sets of parentheses. However, the distributive property is the most straightforward and commonly used method for expanding p.(q+r) or similar expressions.