Question

Find a unit vector in the same direction as the vector A = 4i−2j+ 4k, and another unit vector in the same direction as B = −4i + 3k. Show that the vector sum of these unit vectors bisects the angle between A and B. Hint: Sketch the rhombus having the two unit vectors as adjacent sides.

Answer 1

A unit vector in the same direction as a vector
`v` can be obtained by multiplying
`v` by the reciprocal of its norm. So
`\vec A` and
`\vec B` have corresponding unit vectors
`\vec a` and
`\vec b`,

`\vec a=(\vec A)/(\|\vec A\|)=(4\,\vec\imath-2\,\vec\jmath+4\,\vec k)/(√(4^2+(-2)^2+4^2))=\frac23\,\vec\imath-\frac13\,\vec\jmath+\frac23\,\vec k`

`\vec b=(-4\,\vec\imath+3\,\vec k)/(√((-4)^2+3^2))=-\frac45\,\vec\imath+\frac35\,\vec k`

They have vector sum

`\vec a+\vec b=-\frac2{15}\,\vec\imath-\frac13\,\vec\jmath+(19)/(15)\,\vec k`

Find the angle between
`\vec A` and
`\vec B` using the dot product formula:

`\vec A\cdot\vec B=\|\vec A\|\|\vec B\|\cos\theta`

`-4=30\cos\theta`

`\cos\theta=-\frac2{15}`

`\theta\approx97.66^\circ`

Then find the angle between either
`\vec A` or
`\vec B` and
`\vec a+\vec b`:

`\vec A\cdot(\vec a+\vec b)=\|\vec A\|\|\vec a+\vec b\|\cos\varphi`

`\frac{26}5=2\sqrt{\frac{78}5}\cos\varphi`

`\cos\varphi=\sqrt{(13)/(30)}`

`\varphi\approx48.831^\circ`

(and this is half of
`\theta`)

Answer 2

Answer with Step-by-step explanation:

We are given that

A=4i-2j+4k

B=-4i+3k

`\mid A\mid=√(4^2+(-2)^2+4^2)=6`

`mid B\mid=√(3^2+(-4)^2)=5`

`\hat{A}=(A)/(\mid A\mid)`

`\hat{A}=(4i-2j+4k)/(6)=(2)/(3)i-(1)/(3)j+(2)/(3)k`

`\hat{B}=(-4i+3k)/(5)=(-4)/(5)i+(3)/(5)k`

Sum of unit vectors=
`\hat{A}+\hat{B}`

Sum of unit vectors=
`(2)/(3)i-(1)/(3)j+(2)/(3)k+(-4i+3k)/(5)=(-4)/(5)i+(3)/(5)k`

Sum of unit vectors=
`(-2)/(15)i-(1)/(3)j+(19)/(15)k`

`\mid \hat{A}+\hat{B}\mid=\sqrt{((-2)/(15))^2+((1)/(3))^2+((19)/(15))^2}`

`\mid \hat{A}+\hat{B}\mid=1.32`

`\theta_1=Cos^(-1)((A\cdot B))/(\mid A\mid \mid B\mid)`

`\theta_1=cos^(-1)((-4)/(30))=97.6^(\circ)`

`\theta_2=cos^(-1)(((Sum\;of\;unt\;vectors\cdot A))/(\mid sum\mid \mid A\mid )`

`\theta_2=cos^(-1)((78)/(15\cdot 2\cdot 1.32\cdot 6))=49^(\circ)`

`(1)/(2)\theta_1=(1)/(2)(97.6)=48.8\sim 49^(\circ)=\theta_2`

Hence, proved.