Vectors help!

[suskieanna]

Hello I am struggling with this question. Can anyone help me with this question? I will really appreciate it if you do

3. Points A and B are defined by the position vectors a = 2i - 2j - k and b = 3i + 4k.
b) Find the unit vector which bisects angle AOB.

[S_R_K]

Hello I am struggling with this question. Can anyone help me with this question? I will really appreciate it if you do

3. Points A and B are defined by the position vectors a = 2i - 2j - k and b = 3i + 4k.
b) Find the unit vector which bisects angle AOB.

Use the fact that diagonals of a parallelogram bisect the interior angles.

Can you use the position vectors a and b to find a point C such that OACB is a parallelogram? From there the problem is straightforward.

[suskieanna]

Use the fact that diagonals of a parallelogram bisect the interior angles.

Can you use the position vectors a and b to find a point C such that OACB is a parallelogram? From there the problem is straightforward.

The question first asked to find the unit vector of a and the unit vector of b.

[S_R_K]

The question first asked to find the unit vector of a and the unit vector of b.

I don't think that's obviously helpful, because the sum of two unit vectors is, in general, not a unit vector.

The most straightforward way to do this problem is to just use a and b to find a vector that gives the diagonal of a parallelogram OACB, and then find the unit vector in the same direction.

[AlphaZero]

Use the fact that diagonals of a parallelogram bisect the interior angles.

This is only true for a rhombus.

The diagonals of a parallelogram will always bisect each other, but not necessarily the angles at which they meet.

Note that in the diagram below,  it is not necessarily true that  \(\angle ADE=\angle EDC\),  but we do indeed have  \(\overline{AE}=\overline{EC}\) for example.

The question first asked to find the unit vector of a and the unit vector of b.

This is precisely why the first part of the question asks you to find unit vectors in the direction of  \(\vec{a}\)  and  \(\vec{b}\).

Once you have  \(\hat{a}\)  and  \(\hat{b}\),  you can proceed to use this property of a rhombus to answer the question.

[S_R_K]

Thanks AlphaZero for correcting my error.