What is the value of ||\vec{x}||How to Approach a Vector Projection Problem?

In summary, the question is about proving that \vec{p} is orthogonal to \vec{z}, and that \vec{p} is a scalar multiple of \vec{y}. It also asks to determine the value of ||\vec{x}|| given that ||\vec{p}|| = 6 and ||\vec{z}|| = 8. The key to solving this problem is to use the dot product and the properties of perpendicular vectors, as well as the 3-4-5 right triangle.
  • #1
msimmons
17
0
Problem:
Let [tex]\vec{x}[/tex] and [tex]\vec{y}[/tex] be vectors in Rn and define

[tex]p = \frac{x^Ty}{y^Ty}y[/tex]
and
[tex]z = x - p[/tex]

(a) Show that [tex]\vec{p}\bot\vec{z}[/tex]. Thus [tex]\vec{p}[/tex] is the vector projection of x onto y; that is [tex]\vec{x} = \vec{p} + \vec{z}[/tex], where [tex]\vec{p}[/tex] and [tex]\vec{z}[/tex] are orthogonal components of [tex]\vec{x}[/tex], and [tex]\vec{p}[/tex] is a scalar multiple of [tex]\vec{y}[/tex]

(b) If [tex]||\vec{p}|| = 6[/tex] and [tex]||\vec{z}|| = 8[/tex], determine the value of [tex]||\vec{x}||[/tex]

My problem:
I understand the question, but have no idea how to approach it. Hints?
 
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  • #2
Draw any arbitrary two vectors, draw the projection of one onto the other and stare at it until you realize that [tex] \sqrt{6^2 + 8^2} [/tex] is obvious.
 
  • #3
And then think "3-4-5 right triangle" twice! :rofl:

Your "[itex]x^T y[/itex] is a fancy way of writing the dot product. In my simpler mind, what you really want to prove is that
[tex]\frac{\vec{x}\cdot\vec{y}}{||\vec{x}||} \vec{x}[/tex]
is perpendicular to [itex]\vec{p}- \vec{x}[/itex].

Okay, go ahead and take the dot product:
[tex]\frac{\vec{x}\cdot\vec{y}}{||\vec{x}||} \vec{x}\cdot (\vec{x}- \frac{\vec{x}\dot\vec{y}}{||\vec{x}||} \vec{x})[/itex]
 

1. What is a vector projection?

Vector projection is a mathematical process that involves finding the component of a vector that lies in a specific direction. It is used to break down a vector into its individual components, making it easier to calculate and manipulate.

2. How is vector projection calculated?

Vector projection can be calculated using the dot product formula: projab = (a · b / a · a) * a. This formula takes into account the magnitude and direction of the vectors to determine the projected vector.

3. What is the difference between vector projection and scalar projection?

The main difference between vector projection and scalar projection is that vector projection takes into account both the magnitude and direction of the vectors, while scalar projection only considers the magnitude of the vectors. Vector projection results in a vector, while scalar projection results in a scalar value.

4. What are some real-life applications of vector projection?

Vector projection has various applications in fields such as physics, engineering, and computer graphics. It is used to calculate forces and motion in physics, determine the strength of structural components in engineering, and create realistic 3D graphics in computer graphics.

5. Can vector projection be performed in higher dimensions?

Yes, vector projection can be performed in any number of dimensions. The formula for vector projection remains the same regardless of the dimension, but the calculations may become more complex as the number of dimensions increases.

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