Question

If `|bara|` = 3 and `|barb|` = 4, then the value of λ for which `bara + lambda barb` is perpendicular to `bara - lambda barb` is ______.

Options

• `9/16`

• `3/4`

• `3/2`

• `4/3`

Answer 1

If `|bara|` = 3 and `|barb|` = 4, then the value of λ for which `bara + lambda barb` is perpendicular to `bara - lambda barb` is `bbunderline(3/4)`.

Explanation:

If two vectors are perpendicular, their dot product is 0.

`(bara + lambdabarb) · (bara - lambdabarb) = 0`

Use the distributive property of the dot product:

`bara · bara - lambdabara · barb + lambdabarb · bara - lambda^2barb · barb = 0`

Note that `bara · barb = barb · bara`, so:

`bara · bara - lambda^2barb · barb = 0`

Now plug in magnitudes:

`|bara|^2 = 3^2 = 9, |barb|^2 = 4^2 = 16`

So, 9 − λ² ⋅ 16 = 0

`lambda^2 = 9/16`

`lambda = 3/4`