Question

\(\mathbb{R}^{2}\) Consider the vectors \(\mathbf{I}=\langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle\) and \(\mathbf{J}=\langle-1 / \sqrt{2}, 1 / \sqrt{2}\rangle\). Express \(\mathbf{I}\) and \(\mathbf{J}\) in terms of the usual unit coordinate vectors i and j. Then, write i and \(\mathbf{j}\) in terms of \(\mathbf{I}\) and \(\mathbf{J}\).

Short answer

Question: Express the standard unit coordinate vectors i and j in terms of the given vectors \(\mathbf{I} = \langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle\) and \(\mathbf{J} = \langle -1 / \sqrt{2}, 1 / \sqrt{2}\rangle\). Answer: \(\mathbf{i} = (1/2)\mathbf{I} + (1/2)\mathbf{J}\) and \(\mathbf{j} = (1/2)\mathbf{I} - (1/2)\mathbf{J}\).

Step-by-step solution

Express \(\mathbf{I}\) and \(\mathbf{J}\) in terms of i and j

We are given \(\mathbf{I} = \langle 1 / \sqrt{2}, 1 / \sqrt{2}\rangle\) and \(\mathbf{J} = \langle -1 / \sqrt{2}, 1 / \sqrt{2}\rangle\). To find the linear combinations of i and j that result in these vectors, we have: \(\mathbf{I} = (1/ \sqrt{2})\mathbf{i} + (1 / \sqrt{2})\mathbf{j}\) \(\mathbf{J} = (-1/ \sqrt{2})\mathbf{i} +(1 / \sqrt{2})\mathbf{j}\)

Express i and j in terms of \(\mathbf{I}\) and \(\mathbf{J}\)

Let i and j be linear combinations of \(\mathbf{I}\) and \(\mathbf{J}\) in the following form: \(\mathbf{i} = a\mathbf{I} + b\mathbf{J}\) \(\mathbf{j} = c\mathbf{I} + d\mathbf{J}\) Now, we can use the relationships we found in Step 1 to solve for the coefficients a, b, c, and d. On equating components in the first equation, we get: \(a(1/ \sqrt{2}) + b(-1/ \sqrt{2}) = 1\) \(a(1/ \sqrt{2}) + b(1/ \sqrt{2})= 0\) Similarly, on equating components in the second equation, we get: \(c(1/ \sqrt{2}) + d(-1 / \sqrt{2})= 0\) \(c(1/ \sqrt{2}) + d(1 / \sqrt{2})= 1\) Using the above four equations, we can solve for a, b, c, and d: a = b = 1/2 c = d = 1/2 Finally, we can write i and j in terms of \(\mathbf{I}\) and \(\mathbf{J}\): \(\mathbf{i} = (1/2)\mathbf{I} + (1/2)\mathbf{J}\) \(\mathbf{j} = (1/2)\mathbf{I} - (1/2)\mathbf{J}\)

Key concepts

Understanding Unit Vectors

Unit vectors are fundamental in vector algebra. They help us understand direction without being affected by magnitude. A unit vector has a length of 1 and is usually represented by a hat symbol, like \( \hat{i} \) and \( \hat{j} \). These vectors form the basis for other vectors in two-dimensional space. For instance, any vector \( \mathbf{v} = \langle x, y \rangle \) can be rewritten as a sum of the unit vectors \( \mathbf{v} = x\hat{i} + y\hat{j} \). This representation allows us to express a vector in terms of its components along the x and y axes.
In our example, we have the vectors \( \mathbf{I} \) and \( \mathbf{J} \), which are expressed in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \). By decomposing these vectors, we get:

• \( \mathbf{I} = (1/ \sqrt{2})\mathbf{i} + (1 / \sqrt{2})\mathbf{j} \)
• \( \mathbf{J} = (-1/ \sqrt{2})\mathbf{i} + (1 / \sqrt{2})\mathbf{j} \)

Understanding this expression helps in realizing how unit vectors contribute to forming other vectors.

Demystifying Coordinate Transformation

Coordinate transformation is the process of converting coordinates from one system to another. It allows us to understand and interpret vectors in different orientations or positions. In our exercise, we transform the vector expressions of \( \mathbf{i} \) and \( \mathbf{j} \) in terms of \( \mathbf{I} \) and \( \mathbf{J} \). These transformations require rearranging and solving algebraic equations that define the relationships between different vector systems.
We start with:

• First, by expressing \( \mathbf{i} \) as a combination of \( \mathbf{I} \) and \( \mathbf{J} \): \( \mathbf{i} = a\mathbf{I} + b\mathbf{J} \).
• Second, by expressing \( \mathbf{j} \) similarly: \( \mathbf{j} = c\mathbf{I} + d\mathbf{J} \).

Through solving these, we determine the coefficients needed to express these transformations. Here, they turn out to be \( a = b = 1/2 \) and \( c = d = 1/2 \). This means each unit vector is equally composed of vectors \( \mathbf{I} \) and \( \mathbf{J} \). This transformation highlights how vectors can be redefined through orientation changes.

Exploring Linear Combinations

Linear combinations are used to express vectors as sums of other vectors, scaled by coefficients. This is crucial in understanding how different vector components can be added together to form new vectors. In this exercise, we express vectors \( \mathbf{I} \) and \( \mathbf{J} \) as linear combinations of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), showing the versatility and interchangeability of vector forms.
To achieve this, we use the following combinations:

• \( \mathbf{I} = (1/ \sqrt{2})\mathbf{i} + (1 / \sqrt{2})\mathbf{j} \)
• \( \mathbf{J} = (-1/ \sqrt{2})\mathbf{i} + (1 / \sqrt{2})\mathbf{j} \)

Next, we reversed the roles and expressed \( \mathbf{i} \) and \( \mathbf{j} \) in terms of \( \mathbf{I} \) and \( \mathbf{J} \) through linear combinations:

• \( \mathbf{i} = (1/2)\mathbf{I} + (1/2)\mathbf{J} \)
• \( \mathbf{j} = (1/2)\mathbf{I} - (1/2)\mathbf{J} \)

This approach allows us to see how different vectors can be constructed from a base set. Linear combinations are powerful tools in vector algebra, making it possible to explore vector relationships and dependencies integrally.