Question

Given \(\bar{x}=(-1,2,0,-7), \bar{y}=(0,0,-1,-2),\) and \(\bar{z}=(2,4,-3,-3)\) in \(E^{4}\), express \(\bar{x}, \bar{y},\) and \(\bar{z}\) as linear combinations of the basic unit vectors. Also, compute their absolute values, their inverses, as well as their mutual sums, differences, dot products, and distances. Are any of them orthogonal? Parallel?

Short answer

None of the vectors are orthogonal or parallel.

Step-by-step solution

Express in terms of unit vectors

In a four-dimensional space, the basic unit vectors are typically \(\bar{e}_1 = (1,0,0,0), \bar{e}_2 = (0,1,0,0), \bar{e}_3 = (0,0,1,0), \text{ and } \bar{e}_4 = (0,0,0,1)\). We can express any vector as a linear combination of these basic unit vectors. For \(\bar{x} = (-1,2,0,-7)\): \\(-1\cdot\bar{e}_1 + 2 \cdot \bar{e}_2 + 0 \cdot \bar{e}_3 - 7\cdot\bar{e}_4\).Similarly, \(\bar{y} = (0,0,-1,-2)\) can be expressed as:\(0\cdot\bar{e}_1 + 0\cdot\bar{e}_2 -1\cdot\bar{e}_3 - 2\cdot\bar{e}_4\).And \(\bar{z} = (2,4,-3,-3)\) as:\(2\cdot\bar{e}_1 + 4 \cdot\bar{e}_2 - 3\cdot\bar{e}_3 - 3\cdot\bar{e}_4\).

Compute Absolute Values (Magnitudes)

The magnitude of a vector \(\bar{v} = (v_1, v_2, v_3, v_4)\) is given by the formula:\[\|\bar{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2}.\]For \(\bar{x}\):\(\|\bar{x}\| = \sqrt{(-1)^2 + 2^2 + 0^2 + (-7)^2} = \sqrt{54} = 3\sqrt{6}.\)For \(\bar{y}\):\(\|\bar{y}\| = \sqrt{0^2 + 0^2 + (-1)^2 + (-2)^2} = \sqrt{5}.\)For \(\bar{z}\):\(\|\bar{z}\| = \sqrt{2^2 + 4^2 + (-3)^2 + (-3)^2} = \sqrt{38}.\)

Compute Inverses of Vectors

The inverse of a vector \(\bar{v} = (v_1, v_2, v_3, v_4)\) is simply \(-\bar{v} = (-v_1, -v_2, -v_3, -v_4)\).For \(\bar{x}\):\(-\bar{x} = (1, -2, 0, 7)\).For \(\bar{y}\):\(-\bar{y} = (0, 0, 1, 2)\).For \(\bar{z}\):\(-\bar{z} = (-2, -4, 3, 3)\).

Compute Mutual Sums and Differences

To find the sum or difference of two vectors \(\bar{a} = (a_1, a_2, a_3, a_4)\) and \(\bar{b} = (b_1, b_2, b_3, b_4)\), simply add or subtract them componentwise:\(\bar{x} + \bar{y} = (-1+0, 2+0, 0-1, -7-2) = (-1, 2, -1, -9)\).\(\bar{x} - \bar{y} = (-1-0, 2-0, 0+1, -7+2) = (-1, 2, 1, -5)\).\(\bar{y} + \bar{z} = (0+2, 0+4, -1-3, -2-3) = (2, 4, -4, -5)\).\(\bar{y} - \bar{z} = (0-2, 0-4, -1+3, -2+3) = (-2, -4, 2, 1)\).\(\bar{z} + \bar{x} = (2-1, 4+2, -3+0, -3-7) = (1, 6, -3, -10)\).\(\bar{z} - \bar{x} = (2+1, 4-2, -3-0, -3+7) = (3, 2, -3, 4)\).

Compute Dot Products

The dot product of two vectors \(\bar{a} = (a_1, a_2, a_3, a_4)\) and \(\bar{b} = (b_1, b_2, b_3, b_4)\) is given by:\(\bar{a} \cdot \bar{b} = a_1b_1 + a_2b_2 + a_3b_3 + a_4b_4\).\(\bar{x} \cdot \bar{y} = (-1)\cdot0 + 2\cdot0 + 0\cdot(-1) + (-7)\cdot(-2) = 14\).\(\bar{x} \cdot \bar{z} = (-1)\cdot2 + 2\cdot4 + 0\cdot(-3) + (-7)\cdot(-3) = 25\).\(\bar{y} \cdot \bar{z} = 0\cdot2 + 0\cdot4 + (-1)\cdot(-3) + (-2)\cdot(-3) = 9\).

Compute Distances

The distance between two vectors \(\bar{a}\) and \(\bar{b}\) is given by the magnitude of their difference:\(d(\bar{a},\bar{b}) = \|\bar{a} - \bar{b}\|\).For \(\bar{x} ext{ and } \bar{y}\):\(\|\bar{x} - \bar{y}\| = \sqrt{(-1-0)^2 + (2-0)^2 + (0+1)^2 + (-7+2)^2} = \sqrt{31}\).For \(\bar{y} ext{ and } \bar{z}\):\(\|\bar{y} - \bar{z}\| = \sqrt{(0-2)^2 + (0-4)^2 + (-1+3)^2 + (-2+3)^2} = \sqrt{30}\).For \(\bar{z} ext{ and } \bar{x}\):\(\|\bar{z} - \bar{x}\| = \sqrt{(2+1)^2 + (4-2)^2 + (-3-0)^2 + (-3+7)^2} = \sqrt{38}\).

Check for Orthogonality and Parallelism

Two vectors are orthogonal if their dot product is zero and parallel if one is a scalar multiple of the other.From the dot products calculated:- \(\bar{x} \cdot \bar{y} = 14, \bar{x} \cdot \bar{z} = 25, \bar{y} \cdot \bar{z} = 9\).None of these are zero, so none of the pairs are orthogonal.To check for parallelism, assess if \(\bar{x} \), \(\bar{y}\), or \(\bar{z}\) can be written as scalar multiples of each other. Here, none of the vectors are scalar multiples, so none are parallel.

Key concepts

Vectors

Vectors are fundamental in linear algebra, representing quantities that have both magnitude and direction. They can exist in spaces of any dimension. Here, we are focusing on vectors in a four-dimensional space, denoted as \(E^4\). Understanding vectors involves knowing their components and how they relate to the unit vectors (or basis vectors) of their space.

- A vector can be written as a linear combination of unit vectors. For instance, in \(E^4\), the vector \(\bar{x} = (-1,2,0,-7)\) can be expressed using the unit vectors as \(-1\cdot\bar{e}_1 + 2 \cdot \bar{e}_2 + 0 \cdot \bar{e}_3 - 7\cdot \bar{e}_4\). Here, \(\bar{e}_1, \bar{e}_2, \bar{e}_3, \text{ and } \bar{e}_4\) represent the directions in the four-dimensional space.

Unit vectors have a magnitude of one and point solely in the direction of one of the axes in the vector space. They are the foundational building blocks that allow us to construct any vector in that space.

Dot Product

The dot product, also known as the scalar product, is a way of multiplying two vectors to produce a scalar (a single number). It is useful for understanding angles between vectors and other geometric properties.

The formula for finding the dot product of two vectors \(\bar{a} = (a_1, a_2, a_3, a_4)\) and \(\bar{b} = (b_1, b_2, b_3, b_4)\) is:

• \(\bar{a} \cdot \bar{b} = a_1b_1 + a_2b_2 + a_3b_3 + a_4b_4\)

If the result is zero, the vectors are orthogonal (perpendicular to each other). This concept is especially helpful in determining orthogonality, which is a criteria we analyzed in this exercise.

In the given problem, none of the calculated dot products of \(\bar{x} \cdot \bar{y}\), \(\bar{x} \cdot \bar{z}\), and \(\bar{y} \cdot \bar{z}\) result in zero, indicating that none of the vector pairs are orthogonal.

Vector Magnitude

The magnitude (or length) of a vector is a measure of how long the vector is. In geometric terms, it is the distance from the origin to the point defined by the vector.

To calculate the magnitude of a vector \(\bar{v} = (v_1, v_2, v_3, v_4)\) in four-dimensional space, we use the formula:

• \(\|\bar{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2}\)

Knowing the magnitude of a vector allows us to understand its scale and size, relevant for determining the vector's impact or representation in a particular space.

In the context of this problem, we found the magnitudes of \(\bar{x}\), \(\bar{y}\), and \(\bar{z}\) as \(3\sqrt{6}\), \(\sqrt{5}\), and \(\sqrt{38}\) respectively.

Orthogonality

Orthogonality is a crucial concept in linear algebra, involving the relationship between two vectors. Two vectors are orthogonal if they are perpendicular to each other, meaning their dot product results in zero.

Orthogonality is used in various applications, such as finding the shortest path, minimizing distances, and simplifying calculations in vector spaces. It often signals an independence of direction between the vectors involved.

In this exercise, despite calculating the dot products of the given vectors, none indicated orthogonality, meaning no zero results were found. This insight helps clarify that the spatial relationships between these vectors are not perpendicular, affecting how they interact in the four-dimensional space.