Simple vector/linear algebra question

Hi guys,

I'm currently enrolled in a course in university called Vector Geometry and Linear Algebra. I've never been too strong at math but with some dedication I've always been able to push through. I'm stuck on one type aspect of a question that contains missing values.

The question provides two vectors in $\displaystyle \mathbb{R}^3$, $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$. It provides all three components for $\displaystyle \vec{u}$, but the last component of $\displaystyle \vec{v}$, $\displaystyle z$, is missing. It then asks "For what values of $\displaystyle z$ will the two vectors be: parallel, orthogonal, equal length?"

I understood every other question on the assignment and so far I'm understanding the material, I'm just having a hard time applying the vector methods we've learned to this problem. The missing last component threw me off.

Thanks for any help you can provide, I really appreciate it.

Re: Simple vector/linear algebra question

If all components of both vectors are given, can you determine if the vectors are parallel, orthogonal, or have equal lengths? Do the same thing with z, and in each case you'll get an equation (or two) in z. You need to solve those equations.

Re: Simple vector/linear algebra question

Quote:

Originally Posted by

**sobering** The question provides two vectors in $\displaystyle \mathbb{R}^3$, $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$. It provides all three components for $\displaystyle \vec{u}$, but the last component of $\displaystyle \vec{v}$, $\displaystyle z$, is missing. It then asks "For what values of $\displaystyle z$ will the two vectors be: parallel, orthogonal, equal length?"

You ought post the actual question. Then the explanation is easier to make,

In general: two vectors are parallel if they are multiples of each other.

Two vectors are orthogonal if their dot product is zero.

Same length, $\displaystyle \sqrt{u_x^2+u_y^2+u_z^2}=\sqrt{v_x^2+v_y^2+v_z^2}$

Re: Simple vector/linear algebra question

Hey emakarov and Plato, thanks for your answers. After reading both your posts the solution immediately became obvious. I suppose I was overthinking the question.

I didn't want to post the actual question because I wanted to understand the process, and not just receive an answer. The vectors were $\displaystyle <4, 2, 7>$ and $\displaystyle <2, 1, k>$.

For the orthogonal aspect of the question, here's my solution:

$\displaystyle <4, 2, 7> \cdot <2, 1, k> = 0$

$\displaystyle = 4(2) + 2(1) + 7k = 0$

$\displaystyle k = -10$

Thanks again for all your help.

Re: Simple vector/linear algebra question

Quote:

Originally Posted by

**sobering** Hey emakarov and Plato, thanks for your answers. After reading both your posts the solution immediately became obvious. I suppose I was overthinking the question.

I didn't want to post the actual question because I wanted to understand the process, and not just receive an answer. The vectors were $\displaystyle <4, 2, 7>$ and $\displaystyle <2, 1, k>$.

For the orthogonal aspect of the question, here's my solution:

$\displaystyle <4, 2, 7> \cdot <2, 1, k> = 0$

$\displaystyle = 4(2) + 2(1) + 7k = 0$

$\displaystyle k = -10$

Thanks again for all your help.

Yes, that's correct. To be parallel one must be a multiple of the other- you must have <4. 2, 7>= a<2, 1, k> and so must have 4= 2a, 2= a, 7= ka.

To have the same lenght, as Plato said, you must have $\displaystyle \sqrt{4^2+ 2^2+ 7^2}= \sqrt{2^2+ 1^2+ k^2}$