Question

If the position vectors of the points A (3, 4), B (5, −6) and C (4, −1) are \[\vec{a,}\] \[\vec{b,}\] \[\vec{c}\] respectively, compute \[\vec{a} + 2 \vec{b} - 3 \vec{c}\].

Answer 1

Let \[\vec{a} , \vec{b} , \vec{c}\]  are the position vectors of the points \[A\left( 3, 4 \right)\] \[B\left( 5, - 6 \right)\] and \[C\left( 4, - 1 \right)\] 
Then,
\[\vec{a} = 3 \stackrel\frown{i} + 4 \stackrel\frown{j}\]
\[\vec{b} = 5\stackrel\frown{i} - 6 \stackrel\frown{j}\]
\[\vec{c} = 4 \stackrel\frown{i} - \stackrel\frown{j}\]
Therefore,
\[\vec{a} + 2 \vec{b} - 3 \vec{c} \]
\[ = 3 \stackrel\frown{i} + 4 \stackrel\frown{j} + 2 \left( 5 \stackrel\frown{i} - 6 \stackrel\frown{j} \right) - 3 \left( 4 \stackrel\frown{i} - \stackrel\frown{j} \right)\]
\[ = 3 \stackrel\frown{i} + 4 \stackrel\frown{j} + 10 \stackrel\frown{i} - 12 \stackrel\frown{j} - 12 \stackrel\frown{i} + 3 \stackrel\frown{j} \]
\[ = \stackrel\frown{i} - 5 \stackrel\frown{j}\]