Vectors (dot and cross product worksheet) - educational diagram showing vector operations and mathematical formulas.

It looks like you've uploaded a worksheet on vectors, specifically focusing on dot and cross products. However, there is no specific problem or question visible in the image you provided — it appears to be a set of definitions, explanations, and diagrams about vector operations.

To help you properly, I’ll summarize the key concepts from the worksheet and then explain how to solve typical problems involving dot and cross products, since no specific task was included.

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🔹 Summary of Key Concepts from the Worksheet

#### 1. Vectors
- Quantities with both magnitude and direction.
- Examples: velocity, acceleration, force, momentum.

#### 2. Vector Addition
- To add vectors $ \vec{a} $ and $ \vec{b} $:
- Place the tail of $ \vec{b} $ at the head of $ \vec{a} $.
- The resultant vector $ \vec{a} + \vec{b} $ goes from the tail of $ \vec{a} $ to the head of $ \vec{b} $.

#### 3. Scalar Multiplication
- Multiplying a vector $ \vec{a} $ by a scalar $ k $:
- Magnitude becomes $ |k| \cdot |\vec{a}| $
- Direction remains same if $ k > 0 $, reverses if $ k < 0 $

#### 4. Dot Product (Scalar Product)
- Definition:
$$
\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta
$$
where $ \theta $ is the angle between $ \vec{a} $ and $ \vec{b} $.

- Properties:
- If vectors are parallel: $ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| $
- If vectors are perpendicular: $ \vec{a} \cdot \vec{b} = 0 $
- Commutative: $ \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} $

- In component form (in 3D):
$$
\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z
$$

#### 5. Cross Product (Vector Product)
- Definition:
$$
\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \, \hat{n}
$$
where $ \hat{n} $ is a unit vector perpendicular to both $ \vec{a} $ and $ \vec{b} $, direction determined by the right-hand rule.

- Properties:
- Result is a vector perpendicular to the plane of $ \vec{a} $ and $ \vec{b} $
- Anti-commutative: $ \vec{a} \times \vec{b} = -(\vec{b} \times \vec{a}) $
- If vectors are parallel: $ \vec{a} \times \vec{b} = \vec{0} $
- Magnitude equals area of parallelogram formed by $ \vec{a} $ and $ \vec{b} $

- In component form:
$$
\vec{a} \times \vec{b} =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
a_x & a_y & a_z \\
b_x & b_y & b_z \\
\end{vmatrix}
$$

#### 6. Unit Vectors
- Standard basis vectors:
- $ \hat{i}, \hat{j}, \hat{k} $ along x, y, z axes respectively
- Dot product rules:
$$
\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1 \\
\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0
$$
- Cross product rules:
$$
\hat{i} \times \hat{j} = \hat{k},\quad \hat{j} \times \hat{k} = \hat{i},\quad \hat{k} \times \hat{i} = \hat{j} \\
\hat{j} \times \hat{i} = -\hat{k},\quad \text{etc.}
$$

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🔹 Example Problems You Might Encounter

Since no specific problem was given, here are two common types of questions based on this worksheet:

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✔ Example 1: Dot Product Problem

Problem:
Given $ \vec{a} = 3\hat{i} + 4\hat{j} $, $ \vec{b} = 2\hat{i} - 1\hat{j} $. Find $ \vec{a} \cdot \vec{b} $.

Solution:
$$
\vec{a} \cdot \vec{b} = (3)(2) + (4)(-1) = 6 - 4 = 2
$$

✔ Answer: $ \vec{a} \cdot \vec{b} = 2 $

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✔ Example 2: Cross Product Problem

Problem:
Find $ \vec{a} \times \vec{b} $ where $ \vec{a} = 2\hat{i} + 3\hat{j} $, $ \vec{b} = 1\hat{i} + 4\hat{j} $

Solution:
We can use determinant method:
$$
\vec{a} \times \vec{b} =
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
2 & 3 & 0 \\
1 & 4 & 0 \\
\end{vmatrix}
= \hat{k}(2 \cdot 4 - 3 \cdot 1) = \hat{k}(8 - 3) = 5\hat{k}
$$

✔ Answer: $ \vec{a} \times \vec{b} = 5\hat{k} $

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✔ Example 3: Angle Between Two Vectors Using Dot Product

Problem:
Find the angle between $ \vec{a} = 3\hat{i} + 4\hat{j} $, $ \vec{b} = 1\hat{i} + 2\hat{j} $

Solution:
Step 1: Compute dot product:
$$
\vec{a} \cdot \vec{b} = 3(1) + 4(2) = 3 + 8 = 11
$$

Step 2: Magnitudes:
$$
|\vec{a}| = \sqrt{3^2 + 4^2} = 5,\quad |\vec{b}| = \sqrt{1^2 + 2^2} = \sqrt{5}
$$

Step 3: Use formula:
$$
\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} = \frac{11}{5\sqrt{5}} \Rightarrow \theta = \cos^{-1}\left(\frac{11}{5\sqrt{5}}\right)
$$

✔ Answer: $ \theta \approx \cos^{-1}(0.983) \approx 10^\circ $

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🔍 Final Note

If you have a specific problem (e.g., "Find the dot product of two vectors" or "Calculate the cross product") that was meant to be included but not shown in the image, please describe it or upload the actual question so I can solve it step-by-step.

Alternatively, if your goal was to understand the concepts, I’ve explained everything clearly above with examples!

Let me know how I can assist further!