Consider two general three-dimensional vectors 𝑢̅ and 𝑣̅. If the magnitude of 𝑢̅×𝑣̅ is 𝑘 times larger than the magnitude of 𝑢̅. Prove that the magnitude of 𝑣̅ is greater to or equal to 𝑘. How do you solve it?

1 Answer
May 22, 2018

Please refer to the Explanation for a Proof.

Explanation:

Suppose that, the angle between #vecu and vecv# is #theta#,

where, #0 le theta le pi; vecu, vecv!=vec0#.

Then, we know that, #||vecuxxvecv||=||vecu||||vecv||sintheta#.

But, we are given that, #||vecuxxvecv||=|k|||vecu||; k!=0#.

#:. |k|||vecu||=||vecu||||vecv||sintheta#.

#:." Dividing by "||vecu||!=0," we get, "|k|=||vecv||sintheta...(ast)#.

Now, #0 le sintheta le 1#.

Multiplying by #||vecv|| gt 0," we have, "0 le ||vecv||sintheta le ||vecv||#.

#:." By "(ast), 0 le |k| le ||vecv||#, as desired!