Math Problem Statement

Find the area of the parallelogram that has the vectors as adjacent sides: u = j, v = j + k.

Solution

To find the area of the parallelogram formed by two vectors u⃗\vec{u} and v⃗\vec{v} as adjacent sides, we need to use the formula:

Area=∥u⃗×v⃗∥\text{Area} = \|\vec{u} \times \vec{v}\|

where ∥u⃗×v⃗∥\|\vec{u} \times \vec{v}\| is the magnitude of the cross product of u⃗\vec{u} and v⃗\vec{v}.

From the image:

  • u⃗=j\vec{u} = \mathbf{j}
  • v⃗=j+k\vec{v} = \mathbf{j} + \mathbf{k}

Step 1: Express the vectors in component form

For the basis vectors i\mathbf{i}, j\mathbf{j}, and k\mathbf{k}:

  • u⃗=0i+1j+0k=(0,1,0)\vec{u} = 0\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} = (0, 1, 0)
  • v⃗=0i+1j+1k=(0,1,1)\vec{v} = 0\mathbf{i} + 1\mathbf{j} + 1\mathbf{k} = (0, 1, 1)

Step 2: Calculate the cross product u⃗×v⃗\vec{u} \times \vec{v}

The cross product of u⃗=(0,1,0)\vec{u} = (0, 1, 0) and v⃗=(0,1,1)\vec{v} = (0, 1, 1) is given by:

\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 1 & 0 \\ 0 & 1 & 1 \\ \end{vmatrix}$$ Expanding this determinant: $$\vec{u} \times \vec{v} = \mathbf{i}(1 \cdot 1 - 0 \cdot 1) - \mathbf{j}(0 \cdot 1 - 0 \cdot 1) + \mathbf{k}(0 \cdot 1 - 1 \cdot 0)$$ $$= \mathbf{i}(1) - \mathbf{j}(0) + \mathbf{k}(0) = \mathbf{i}$$ So, $$\vec{u} \times \vec{v} = (1, 0, 0)$$. ### Step 3: Find the magnitude of the cross product The magnitude of $$(1, 0, 0)$$ is: $$\|\vec{u} \times \vec{v}\| = \sqrt{1^2 + 0^2 + 0^2} = \sqrt{1} = 1$$ ### Conclusion The area of the parallelogram is: $$\text{Area} = 1$$ Would you like further details on any steps or have additional questions? --- ### Follow-up Questions: 1. How do we calculate the cross product for vectors in general? 2. Why does the cross product's magnitude represent the area of a parallelogram? 3. What are the conditions under which two vectors are parallel? 4. How does the direction of the cross product relate to the right-hand rule? 5. Can the area ever be negative in this type of calculation? ### Tip: When working with cross products, remember that only three-dimensional vectors can produce a meaningful cross product.

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Math Problem Analysis

Mathematical Concepts

Vector Operations
Cross Product
Area of a Parallelogram

Formulas

Area = |u × v|
Cross Product Formula for Determinants

Theorems

Properties of Cross Product

Suitable Grade Level

Grades 11-12

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