Question

Find a vector 3 units long in the opposite direction from v = 〈3, 1,-4).

Answer 1

Unit vector along the direction v = <3,1,-4> is :

`\hat{v} = (3i + j -4k)/(√(3^2 + 1^2 + 4^2))\\\\\hat{v} = (3i + j -4k)/(5.1)`

So, unit vector opposing the
`\hat{v}` is :

`\hat{v'} = -\hat{v}\\\\\hat{v'} = -( (3i + j -4k)/(5.1))\\\\\hat{v'} = (-3i - j +4k)/(5.1)`

so, vector of magnitude 3 units in opposite direction from v is :

`\vec{V} = 3\hat{v'}\\\\\vec{V} = (3)/(5.1)( -3i -j+4k)`

Hence, this is the required solution.