Question

a, b, c are three vectors of equal magnitudes and each of them is inclined at an angle of 60° to the others. If |a + b + c| = √6, then find |a|.

Answer 1

a ,b , c are three vectors of equal in magnitudes.

so, |a| = |b| = |c| .......(1)

now, |a + b + c | = √{(a + b + c).(a + b + c)}

= √{a.a + a.b + a.c + b.a + b.b + b.c + c.a + c.b + c.c}

= √{|a|² + |b|² + |c|² + 2(a.b + b.c + c.a)}

= √{|a|² + |b|² + |c|² + 2(|a|.|b|cos60° + |b|.|c|cos60° + |c|.|a|cos60°}

= √{|a|² + |b|² + |c|² + |a|.|b| + |b|.|c| + |c|.|a|}

from equation (1),

= √{|a|² + |a|² + |a|² + |a|.|a| + |a|.|a| + |a|.|a|}

= √{3|a|² + 3|a|²}

= √6|a|

now, |a + b + c| = √6 = √6|a|

hence, |a| = 1