Let \[\vec{a} = 4 \vec{i} + 3 \vec{j} \text { and } \vec{b} = 3 \vec{i} + 4 \vec{j}\]. Find the magnitudes of (a) \[\vec{a}\] , (b) \[\vec{b}\] ,(c) \[\vec{a} + \vec{b} \text { and }\] (d) \[\vec{a} - \vec{b}\].
Solution
Given: \[\vec{a} = 4 \vec{i} + 3 \vec{j} \text { and } \vec{b} = 3 \vec{i} + 4 \vec{j}\]
(a) Magnitude of \[\vec{a}\] is given by \[\left| \vec{a} \right| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = 5\]
Magnitude of \[\vec{b}\] is given by \[\left| \vec{b} \right| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5\]
(c) \[\vec{a} + \vec{b} = (4 \hat {i} + 3 \hat {j} ) + (3 \hat { i} + 4 \hat { j} ) = (7 \hat { i} + 7 \hat {j} )\]
∴ Magnitude of vector \[\vec{a} + \vec{b}\] is given by \[\left| \vec{a} + \vec{b} \right| = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}\]
(d) \[\vec{a} - \vec{b} = \left( 4 \vec{i} + 3 \vec{j} \right) - \left( 3 \vec{i} + 4 \vec{j} \right) = \vec{i} - \vec{j}\]
∴ Magnitude of vector \[\vec{a} - \vec{b}\] is given by \[\left| \vec{a} - \vec{b} \right| = \sqrt{\left( 1 \right)^2 + \left( - 1 \right)^2} = \sqrt{2}\]
