Linear Algebra - forming an acute angle

[Leo79]

Hi,

I'm having trouble finding a scalar to suit the equation. It is my understanding that the if the dot product of x+cy • z = 0 or more, it will give an acute angle but I am unsure how to answer question 2c without trial and error.

Any help would be much appreciated,

Leo.

 

[pka]

If \(\theta\) is the angle between \(\vec{v}~\&~\vec{u}\) then \(\vec{v}\cdot\vec{u}=\|\vec{v}\|\,\|\vec{u}\|\cos(\theta)\).
Thus for \(\theta\) to be acute, the dot product must be positive.
So what \(c\) makes
\(\left[ {\begin{array}{*{20}{c}}
{3 - 2c} \\
{ - 1 + 7c} \\
{2 + 3c}
\end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}}
{13} \\
{ - 17} \\
0
\end{array}} \right] > 0\)

 

[Leo79]

If \(\theta\) is the angle between \(\vec{v}~\&~\vec{u}\) then \(\vec{v}\cdot\vec{u}=\|\vec{v}\|\,\|\vec{u}\|\cos(\theta)\).
Thus for \(\theta\) to be acute, the dot product must be positive.
So what \(c\) makes
\(\left[ {\begin{array}{*{20}{c}}
{3 - 2c} \\
{ - 1 + 7c} \\
{2 + 3c}
\end{array}} \right] \cdot \left[ {\begin{array}{*{20}{c}}
{13} \\
{ - 17} \\
0
\end{array}} \right] > 0\)

Hi, yes -I've got to this point but am unsure of how to solve for c from here?

 

[pka]

Hi, yes -I've got to this point but am unsure of how to solve for c from here?

What is the dot product of those two vectors?

 

[Leo79]

What is the dot product of those two vectors?

Here's my answer - I still get an acute answer by substituting it back in

to answer this question I am trying to find c, correct?

 

[pka]

I wish that I could read you post, but alas not luck.
The dot product is \(56-145c\).
You want to solve \(56-145c>0\).

 

[Leo79]

I wish that I could read you post, but alas not luck.
The dot product is \(56-145c\).
You want to solve \(56-145c>0\).

Okay so I've got my answer as c < -56/145.
Firstly is this correct? and secondly is my answer to the question "any value of c less than -56/145 will form an acute angle with z"? If not could I have some help in wording it correctly?

 

[pka]

Okay so I've got my answer as c < -56/145.
Firstly is this correct? and secondly is my answer to the question "any value of c less than -56/145 will form an acute angle with z"? If not could I have some help in wording it correctly?

You need to review basic rules of algebra. It is \(c<\dfrac{56}{145}\).