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Chapter 3: Problem 1

Find the lengths and the inner product of \(x=(1,4,0,2)\) and \(y=(2,-2,1,3)\).

Short Answer

Expert verified
The lengths of vectors x and y are \( \sqrt{21} \) and \( \sqrt{18} \), and their inner product is 0.

Step by step solution

01

Calculate the length of vector x

To find the length (or magnitude) of the vector \( x = (1, 4, 0, 2) \), use the formula for the Euclidean norm: \[ \|x\| = \sqrt{x_1^2 + x_2^2 + x_3^2 + x_4^2} \] Substitute the values: \( \|x\| = \sqrt{1^2 + 4^2 + 0^2 + 2^2} = \sqrt{1 + 16 + 0 + 4} = \sqrt{21} \). So, the length of vector \( x \) is \( \sqrt{21} \).
02

Calculate the length of vector y

Next, calculate the magnitude of \( y = (2, -2, 1, 3) \) using the same formula: \[ \|y\| = \sqrt{y_1^2 + y_2^2 + y_3^2 + y_4^2} \] Substitute the values: \( \|y\| = \sqrt{2^2 + (-2)^2 + 1^2 + 3^2} = \sqrt{4 + 4 + 1 + 9} = \sqrt{18} \). Thus, the length of vector \( y \) is \( \sqrt{18} \).
03

Find the inner product of vectors x and y

To find the inner product (or dot product) of vectors \( x \) and \( y \), use the formula: \[ x \cdot y = x_1y_1 + x_2y_2 + x_3y_3 + x_4y_4 \] Substitute the values: \( x \cdot y = 1 \cdot 2 + 4 \cdot (-2) + 0 \cdot 1 + 2 \cdot 3 = 2 - 8 + 0 + 6 = 0 \). The inner product of vectors \( x \) and \( y \) is \( 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Euclidean norm
The Euclidean norm of a vector is a measure of its length in Euclidean space. Think of it as the distance from the point represented by the vector to the origin in a multi-dimensional space. To compute it, use the formula:\[ \|x\| = \sqrt{x_1^2 + x_2^2 + ... + x_n^2} \]where \( x_1, x_2, ..., x_n \) are components of the vector. Imagine if our vector points were on a graph. The Euclidean norm would be the straight line distance from these points to the origin.
  • The Euclidean norm is sometimes called the L2 norm.
  • It gives the length of a vector in any dimension.
  • Note that the result is always a non-negative value.
In many practical problems, finding the Euclidean norm helps to understand the vector's "size" or "strength" in a given space.
Vector magnitude
The vector magnitude is another way to describe the length of a vector. Essentially, it's the same as the Euclidean norm, measuring how long or how big the vector is.To find the magnitude of a vector \( x = (a, b, c, ...) \), you use:\[ \|x\| = \sqrt{a^2 + b^2 + c^2 + ...} \]
  • This value describes the distance from the vector to the origin in geometric terms.
  • Magnitude becomes especially important in physics and engineering, where it's used to calculate force, velocity, and acceleration.
  • Understanding magnitude helps when comparing vectors; a larger magnitude means a longer vector.
For a vector in three-dimensional space, picture it as an arrow extending from the origin to a point in space. The magnitude is the length of this arrow.
Dot product
The dot product, also known as the inner product, is a way of multiplying two vectors to get a scalar (a single number). It combines corresponding components and sums them up:\[ x \cdot y = x_1y_1 + x_2y_2 + ... + x_ny_n \]This equation calculates the product between two vectors \( x \) and \( y \). The result tells us something about the directional relationship between the two vectors.
  • If the dot product is positive, it suggests that the vectors point in a similar direction.
  • A negative dot product indicates that they point in opposite directions.
  • If the dot product is zero, as in the given exercise, the vectors are perpendicular or orthogonal to each other.
Understanding the dot product is fundamental in many fields such as computer graphics, physics, and machine learning, as it can inform angle calculations and projections between vectors.

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Most popular questions from this chapter

Find the best straight-line fit to the following measurements, and sketch your solution: $$ \begin{aligned} &y=2 \text { at } t=-1, \quad y=0 \quad \text { at } \quad t=0 \\ &y=-3 \text { at } t=1, \quad y=-5 \quad \text { at } \quad t=2 \end{aligned} $$

Find the projection matrix \(P\) onto the space spanned by \(a_{1}=(1,0,1)\) and \(a_{2}=\) \((1,1,-1)\)

If \(P\) is the projection matrix onto a line in the \(x-y\) plane, draw a figure to describe the effect of the "reflection matrix \(^{\prime \prime} H=I-2 P\). Explain both geometrically and algebraically why \(H^{2}=I\).

Let \(\mathbf{S}\) be the subspace of \(\mathbf{R}^{4}\) containing all vectors with \(x_{1}+x_{2}+x_{3}+x_{4}=0\). Find a basis for the space \(S^{\perp}\), containing all vectors orthogonal to \(S\).

How do we know that the \(i\) th row of an invertible matrix \(B\) is orthogonal to the \(j\) th column of \(B^{-1}\), if \(i \neq j\) ?
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