Mathematics

Mathematics

Problem

For vectors a,b,c\mathbf{a},\mathbf{b},\mathbf{c} it is given that a⋅b=2,  a⋅c=4,  b⋅c=−1\mathbf{a}\cdot\mathbf{b}=2,\;\mathbf{a}\cdot\mathbf{c}=4,\;\mathbf{b}\cdot\mathbf{c}=-1 and ∣b∣2=5|\mathbf{b}|^{2}=5. Find the value of (a+2b)⋅(b−c)(\mathbf{a}+2\mathbf{b})\cdot(\mathbf{b}-\mathbf{c}). a) 6 b) 12 c) 10 d) 8
Intuitive explanation

The key is to expand the dot product of the two vector expressions using the distributive property, then substitute the given dot products. Treat (a+2b)⋅(b−c)(\mathbf{a}+2\mathbf{b})\cdot(\mathbf{b}-\mathbf{c}) like multiplying two binomials—each term in the first parenthesis dots with each term in the second—and you'll get a sum of four dot products that you already know.

Intuitive explanation

The key is to expand the dot product of the two vector expressions using the distributive property, then substitute the given dot products. Treat (a+2b)⋅(b−c)(\mathbf{a}+2\mathbf{b})\cdot(\mathbf{b}-\mathbf{c}) like multiplying two binomials—each term in the first parenthesis dots with each term in the second—and you'll get a sum of four dot products that you already know.

Browse the problem library