Saved Bookmarks
| 1. |
If ` vec a , vec b , vec c`are mutually perpenedicularvectors of equal magnitudes, show thatthe vector ` vec a+ vec b+ vec c`is equally inclined to ` vec a , vec b ,a n d vec cdot` |
|
Answer» Let `|vec(a)|=|vec(b)| = |vec(c )|=a` Given that, `vec(a).vec(b) = vec(b).vec(c) = vec(c).vec(a) =0` Now `(vec(a) + vec(b) + vec(c ))^(2)` `=|vec(a)|^(2) + |vec(b)|^(2) + |vec(c )|^(2) + 2(vec(a).vec(b) + vec(b).vec(c ) + vec( c) .vec(a))` `=a^(2) + a^(2) +a ^(2) +0=3a^(2)` `implies |vec(a)+vec(b)+vec(c)|=sqrt(3)a` Let `theta` be the angle between `vec(a)` and `(vec(a) + vec(b) + vec(c ))` `therefore vec(a). (vec(a) + vec(b) + vec(c )) = |vec(a)||vec(a)+vec(b)+vec(c )|costheta_(1)` `implies vec(a) .vec(a)+vec(a).vec(b) + vec(a).vec(c ) = a(asqrt(3))costheta_(1)` `implies a^(2) + 0 + 0 = a^(2) sqrt(3) cos theta_(1)` `implies cos theta_(1) = (a^(2))/(a^(2)sqrt(3)) = (1)/(sqrt(3))` `implies theta_(1) = cos^(-1)((1)/(sqrt(3)))` Similarly, angle between `vec(b)` and `vec(a) + vec(b) + vec(c) = cos^(-1) .(1)/(sqrt(3))` and angle between `vec(c ) and vec(a) + vec(b) + vec(c ) = cos^(-1).(1)/(sqrt(3))`. Therefore, `vec(a) + vec(b) + vec(c )` makes equal angles with the vectors `vec(a),vec(b)` and `vec(c )`. Hence Proved. |
|