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Worked solution

Practise componentwise scalar products

If a=i+4j−k\mathbf a=\mathbf i+4\mathbf j-\mathbf k, b=i−j+3k\mathbf b=\mathbf i-\mathbf j+3\mathbf k and c=2i+j−5k\mathbf c=2\mathbf i+\mathbf j-5\mathbf k, find (i) a⋅c\mathbf a\cdot\mathbf c and (ii) (2a+3b)⋅c(2\mathbf a+3\mathbf b)\cdot\mathbf c.

CJC · Vectors Basic · Self-Practice 2

(i) a dot c

Answer

1111.

Multiply matching components

a⋅c=1(2)+4(1)+(−1)(−5)=11\begin{aligned}&\mathbf a\cdot\mathbf c\\ & =1(2)+4(1)+(-1)(-5)\\ & =11\end{aligned}

The product of the two negative third components is positive.

How did this part go?

(ii) A linear combination dot c

Answer

−20-20.

Form the combination then dot

2a+3b=(557),(2a+3b)⋅c=5(2)+5(1)+7(−5)=−202\mathbf a+3\mathbf b=\begin{pmatrix}5\\5\\7\end{pmatrix},\qquad(2\mathbf a+3\mathbf b)\cdot\mathbf c=5(2)+5(1)+7(-5)=-20

These explanations are authored supplements to the printed short answers.

How did this part go?

Solution notes

Both printed final answers are retained. The source supplies no full working for this self-practice; the steps are authored.