https://file.aiquickdraw.com/mathbot/file/172890821559077b4zi8f.jpg

Let's go through the problem step by step:

### (i) Calculate the magnitudes \(|\mathbf{v}_1|\) and \(|\mathbf{v}_2|\):

Given vectors:
\[
\mathbf{v}_1 = 5\hat{i} + 7\hat{j}, \quad \mathbf{v}_2 = 3\hat{i} + 4\hat{j}
\]

To find the magnitudes, use the formula for the magnitude of a vector:
\[
|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}
\]

- For \(\mathbf{v}_1 = 5\hat{i} + 7\hat{j}\):
\[
|\mathbf{v}_1| = \sqrt{5^2 + 7^2} = \sqrt{25 + 49} = \sqrt{74}
\]

- For \(\mathbf{v}_2 = 3\hat{i} + 4\hat{j}\):
\[
|\mathbf{v}_2| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

Thus:
\[
|\mathbf{v}_1| = \sqrt{74}, \quad |\mathbf{v}_2| = 5
\]

---

### (ii) Find the scalar (dot) product, \(\mathbf{v}_d = \mathbf{v}_1 \cdot \mathbf{v}_2\), and use it to find the angle between \(\mathbf{v}_1\) and \(\mathbf{v}_2\):

The dot product formula is:
\[
\mathbf{v}_1 \cdot \mathbf{v}_2 = v_{1x} \cdot v_{2x} + v_{1y} \cdot v_{2y}
\]

- For \(\mathbf{v}_1 = 5\hat{i} + 7\hat{j}\) and \(\mathbf{v}_2 = 3\hat{i} + 4\hat{j}\):
\[
\mathbf{v}_1 \cdot \mathbf{v}_2 = (5)(3) + (7)(4) = 15 + 28 = 43
\]

The angle \(\theta\) between \(\mathbf{v}_1\) and \(\mathbf{v}_2\) can be found using the dot product formula:
\[
\cos(\theta) = \frac{\mathbf{v}_1 \cdot \mathbf{v}_2}{|\mathbf{v}_1| |\mathbf{v}_2|}
\]
Substitute values:
\[
\cos(\theta) = \frac{43}{\sqrt{74} \times 5}
\]
\[
\cos(\theta) = \frac{43}{5\sqrt{74}} \quad \Rightarrow \quad \theta = \cos^{-1} \left(\frac{43}{5\sqrt{74}}\right)
\]

---

### (iii) Find the vector (cross) product, \(\mathbf{v}_c = \mathbf{v}_1 \times \mathbf{v}_2\), and compute \(\mathbf{v}_c \cdot \mathbf{v}_1\) and \(\mathbf{v}_c \cdot \mathbf{v}_2\):

For two vectors in 2D, the cross product gives a scalar result in the \(k\)-direction (z-axis in 3D). The formula for 2D vectors is:
\[
\mathbf{v}_1 \times \mathbf{v}_2 = (v_{1x}v_{2y} - v_{1y}v_{2x})\hat{k}
\]

- For \(\mathbf{v}_1 = 5\hat{i} + 7\hat{j}\) and \(\mathbf{v}_2 = 3\hat{i} + 4\hat{j}\):
\[
\mathbf{v}_1 \times \mathbf{v}_2 = (5)(4) - (7)(3) = 20 - 21 = -1\hat{k}
\]

Thus, \(\mathbf{v}_c = -\hat{k}\).

Now compute the dot products:
\[
\mathbf{v}_c \cdot \mathbf{v}_1 = 0, \quad \mathbf{v}_c \cdot \mathbf{v}_2 = 0
\]
(since \(\mathbf{v}_c\) is perpendicular to both \(\mathbf{v}_1\) and \(\mathbf{v}_2\)).

---

### (iv) Evaluate the magnitude of \(|\mathbf{v}_1 \times \mathbf{v}_2|\) and use it to find the angle between \(\mathbf{v}_1\) and \(\mathbf{v}_2\):

The magnitude of the cross product in 2D is simply the absolute value of the scalar result:
\[
|\mathbf{v}_1 \times \mathbf{v}_2| = | -1 | = 1
\]

Now, the sine of the angle \(\theta\) between the vectors is given by:
\[
|\mathbf{v}_1 \times \mathbf{v}_2| = |\mathbf{v}_1| |\mathbf{v}_2| \sin(\theta)
\]
Substitute values:
\[
1 = \sqrt{74} \times 5 \times \sin(\theta)
\]
\[
\sin(\theta) = \frac{1}{5\sqrt{74}}
\]

Thus, \(\theta = \sin^{-1} \left(\frac{1}{5\sqrt{74}}\right)\).

---

Would you like more details or explanations for any part? Here are some questions to deepen understanding:

1. What is the significance of the scalar (dot) product in vector analysis?
2. How can the angle between two vectors help in physical interpretations?
3. Why is the cross product of two 2D vectors in the \(\hat{k}\)-direction (z-axis)?
4. Can the dot product of the cross product with the original vectors ever be non-zero in 2D?
5. How would the magnitudes of the vectors affect the angle between them?

**Tip:** Remember that the cross product in 2D always results in a vector perpendicular to the plane defined by the two original vectors.

1.4*) Consider the two vectors, v1 = 5î + 7ĵ and v2 = 3î + 4ĵ. (i) Calculate the magnitudes |v1| and |v2|? (ii) Find the scalar (dot) product, vd = v1 ⋅ v2 and use it and your answer to part (1) to obtain the angle between v1 and v2. (iii) Find the vector (cross) product, vc = v1 × v2 and find the value of vc ⋅ v1 and vc ⋅ v2. (iv) Evaluate the magnitude of |v1 × v2|. Use the result and your answer to part (1) to find the angle between v1 and v2. Compare your answer with that from part (2).

Learn how to calculate the magnitudes of two vectors, find their dot and cross products, and determine the angle between them using both methods. This problem covers core vector operations, including formulas for magnitude, dot product, and cross product. Suitable for undergraduate students.

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