7.1 Q-2

Question Statement

Find the magnitude of the vector U for the following cases:

(i) $\underline{U} = 2 \underline{\mathbf{i}} - 7 \underline{\mathbf{j}}$
(ii) $\underline{U} = \underline{\mathbf{i}} + \underline{\mathbf{j}}$
(iii) $\underline{U} = [3, -4]$

---

Background and Explanation

To calculate the magnitude of a vector, we use the Pythagorean theorem. For a vector U in two dimensions, the magnitude is given by: $|\underline{\mathbf{U}}| = \sqrt{x^2 + y^2}$

Where $x$ and $y$ are the components of the vector along the x and y axes, respectively. This formula is essentially the distance from the origin to the point $(x, y)$ in the 2D plane.

---

Solution

(i) $\underline{U} = 2 \underline{\mathbf{i}} - 7 \underline{\mathbf{j}}$

1. Identify the components of the vector:
   The vector $\underline{U}$ has components $x = 2$ and $y = -7$.

2. Apply the magnitude formula:
   $|\underline{\mathbf{U}}| = \sqrt{(2)^2 + (-7)^2}$

3. Simplify the calculation:
   $|\underline{\mathbf{U}}| = \sqrt{4 + 49} = \sqrt{53}$

So, the magnitude of the vector is:
$|\underline{\mathbf{U}}| = \sqrt{53}$

---

(ii) $\underline{U} = \underline{\mathbf{i}} + \underline{\mathbf{j}}$

1. Identify the components of the vector:
   The vector $\underline{U}$ has components $x = 1$ and $y = 1$.

2. Apply the magnitude formula:
   $|\underline{\mathbf{U}}| = \sqrt{(1)^2 + (1)^2}$

3. Simplify the calculation:
   $|\underline{\mathbf{U}}| = \sqrt{1 + 1} = \sqrt{2}$

So, the magnitude of the vector is:
$|\underline{\mathbf{U}}| = \sqrt{2}$

---

(iii) $\underline{U} = [3, -4]$

1. Identify the components of the vector:
   The vector $\underline{U}$ has components $x = 3$ and $y = -4$.

2. Apply the magnitude formula:
   $|\underline{\mathbf{U}}| = \sqrt{(3)^2 + (-4)^2}$

3. Simplify the calculation:
   $|\underline{\mathbf{U}}| = \sqrt{9 + 16} = \sqrt{25}$

4. Take the square root:
   $|\underline{\mathbf{U}}| = 5$

So, the magnitude of the vector is:
$|\underline{\mathbf{U}}| = 5$

---

Key Formulas or Methods Used

• Magnitude of a 2D vector:
  $|\underline{\mathbf{U}}| = \sqrt{x^2 + y^2}$
  Where $x$ and $y$ are the components of the vector along the x and y axes.

---

Summary of Steps

1. Identify the components $x$ and $y$ of the vector U.
2. Apply the magnitude formula:
   $|\underline{\mathbf{U}}| = \sqrt{x^2 + y^2}$
3. Simplify the expression to get the magnitude of the vector.