Two vectors are given by $$ \vec{a}=1.8 \hat{i}+1.6 \hat{j} $$ and $$ \vec{b}=8.8 \hat{i}+7.2 \hat{j} $$. Find (a) $$ |\vec{a} imes \vec{b}|,(\mathrm{b}) \vec{a} \cdot \vec{b},(\mathrm{c})(\vec{a}+\vec{b}) \cdot \vec{b} $$, and (d) the component of $$ \vec{a} $$ along the direction of $$ \vec{b} $$ ?

Question

Two vectors are given by $$ \vec{a}=1.8 \hat{i}+1.6 \hat{j} $$ and $$ \vec{b}=8.8 \hat{i}+7.2 \hat{j} $$. Find (a) $$ |\vec{a} 	imes \vec{b}|,(\mathrm{b}) \vec{a} \cdot \vec{b},(\mathrm{c})(\vec{a}+\vec{b}) \cdot \vec{b} $$, and (d) the component of $$ \vec{a} $$ along the direction of $$ \vec{b} $$ ?

UpStudy AI · Accepted Answer

Let's analyze each part step-by-step using the given vectors:

$$
\vec{a} = 1.8\, \hat{i} + 1.6\, \hat{j} \quad \text{and} \quad \vec{b} = 8.8\, \hat{i} + 7.2\, \hat{j}
$$

### (a) Magnitude of the Cross Product $$ |\vec{a} \times \vec{b}| $$

In two dimensions, the cross product of $$\vec{a}$$ and $$\vec{b}$$ is a scalar given by:

$$
\vec{a} \times \vec{b} = a_x b_y - a_y b_x
$$

Substituting the given components:

$$
\vec{a} \times \vec{b} = (1.8)(7.2) - (1.6)(8.8) = 12.96 - 14.08 = -1.12
$$

The magnitude is:

$$
|\vec{a} \times \vec{b}| = | -1.12 | = 1.12
$$

**Answer:** $$ |\vec{a} \times \vec{b}| = 1.12 $$

---

### (b) Dot Product $$ \vec{a} \cdot \vec{b} $$

The dot product is calculated as:

$$
\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y
$$

Substituting the given components:

$$
\vec{a} \cdot \vec{b} = (1.8)(8.8) + (1.6)(7.2) = 15.84 + 11.52 = 27.36
$$

**Answer:** $$ \vec{a} \cdot \vec{b} = 27.36 $$

---

### (c) Dot Product $$ (\vec{a} + \vec{b}) \cdot \vec{b} $$

First, find $$\vec{a} + \vec{b}$$:

$$
\vec{a} + \vec{b} = (1.8 + 8.8)\, \hat{i} + (1.6 + 7.2)\, \hat{j} = 10.6\, \hat{i} + 8.8\, \hat{j}
$$

Now, calculate the dot product:

$$
(\vec{a} + \vec{b}) \cdot \vec{b} = (10.6)(8.8) + (8.8)(7.2) = 93.28 + 63.36 = 156.64
$$

**Answer:** $$ (\vec{a} + \vec{b}) \cdot \vec{b} = 156.64 $$

---

### (d) Component of $$ \vec{a} $$ Along the Direction of $$ \vec{b} $$

The component of $$\vec{a}$$ along $$\vec{b}$$ is given by:

$$
\text{Component} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}
$$

First, compute $$ |\vec{b}| $$:

$$
|\vec{b}| = \sqrt{8.8^2 + 7.2^2} = \sqrt{77.44 + 51.84} = \sqrt{129.28} \approx 11.37
$$

Now, use the previously calculated dot product:

$$
\text{Component} = \frac{27.36}{11.37} \approx 2.406
$$

Rounding to two decimal places:

$$
\text{Component} \approx 2.41
$$

**Answer:** The component of $$ \vec{a} $$ along $$ \vec{b} $$ is approximately 2.41 units.
(a) $$ |\vec{a} \times \vec{b}| = 1.12 $$ (b) $$ \vec{a} \cdot \vec{b} = 27.36 $$ (c) $$ (\vec{a} + \vec{b}) \cdot \vec{b} = 156.64 $$ (d) The component of $$ \vec{a} $$ along $$ \vec{b} $$ is approximately 2.41 units.

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