Q.Evaluate the product $(3\vec{a}-5\vec{b}) \cdot (2\vec{a}+7\vec{b}).$

Concept: Vector Magnitude Properties — the dot product distributes like ordinary multiplication, and $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$.

Step 1: Expand using the distributive property:

Step 2: Simplify each term using $\vec{a}\cdot\vec{a}=|\vec{a}|^2$, $\vec{b}\cdot\vec{b}=|\vec{b}|^2$, and $\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$: …

When you see a product of two vector expressions like this, the instinct should be to treat the dot product just like an algebraic multiplication — but with one crucial difference: the dot product is commutative (order doesn’t matter) but it’s not associative with scalars in the same way. Here, the scalars (3, -5, 2, 7) just multiply through normally.

The real work is in expanding carefully and then grouping like terms. Let’s do it step by step.

1. Expand using the distributive property The dot product distributes over addition, just like ordinary multiplication:

2. Pull out the scalar coefficients For any scalars $m, n$ and vectors $\vec{u}, \vec{v}$, we have $(m\vec{u}) \cdot (n\vec{v}) = mn (\vec{u} \cdot \vec{v})$. So:

That simplifies to:

3. Use commutativity of the dot product The dot product is commutative: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$. So the two middle terms can be combined:

4. Rewrite in standard notation …

Method: Expanding a Dot Product of Two Linear Combinations

Use this whenever you must evaluate a product such as $(3\vec{a}-5\vec{b})\cdot(2\vec{a}+7\vec{b})$ where the individual magnitudes/dot product may or may not be given.

Steps

Step 1: Distribute like a FOIL expansion.

The dot product is distributive, so multiply every term of the first bracket by every term of the second:

Scalars pull out: $(m\vec u)\cdot(n\vec v)=mn(\vec u\cdot\vec v)$.

Step 2: Use commutativity to merge the middle terms.

Since $\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$, combine $21(\vec a\cdot\vec b)-10(\vec a\cdot\vec b)=11(\vec a\cdot\vec b)$. …

Common Mistakes

Mistake 1: Mishandling the sign when merging the middle terms.

Why it's wrong: the terms are $+21(\vec a\cdot\vec b)$ and $-10(\vec a\cdot\vec b)$, so they combine to $+11(\vec a\cdot\vec b)$, not $31$. Correct approach: respect the sign from the $-5\vec b$ factor before adding.

Mistake 2: Treating $\vec{a}\cdot\vec{b}$ and $\vec{b}\cdot\vec{a}$ as different.

Why it's wrong: the dot product is commutative, so they are the same term and can be merged. Correct approach: use $\vec a\cdot\vec b=\vec b\cdot\vec a$ to collect the cross terms. …