In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
||w - u||

Verified step by step guidance

1. Identify the vectors given: \( \mathbf{u} = 2\mathbf{i} - 5\mathbf{j} \), \( \mathbf{w} = -\mathbf{i} - 6\mathbf{j} \).
2. Calculate the vector difference \( \mathbf{w} - \mathbf{u} \) by subtracting the components of \( \mathbf{u} \) from \( \mathbf{w} \):
3. \[ \mathbf{w} - \mathbf{u} = (-1 - 2)\mathbf{i} + (-6 - (-5))\mathbf{j} \]
4. Simplify the components to get the resulting vector:
5. \[ \mathbf{w} - \mathbf{u} = (-3)\mathbf{i} + (-1)\mathbf{j} \]
6. Find the magnitude (or norm) of the vector \( \mathbf{w} - \mathbf{u} \) using the formula for the length of a vector in 2D:
7. \[ ||\mathbf{w} - \mathbf{u}|| = \sqrt{(-3)^2 + (-1)^2} \]

Key Concepts

Vector Subtraction

Vector subtraction involves finding the difference between two vectors by subtracting their corresponding components. For vectors u and w, w - u is computed by subtracting each component of u from the corresponding component of w, resulting in a new vector.

Magnitude (Norm) of a Vector

The magnitude of a vector is its length in the coordinate plane, calculated using the Pythagorean theorem. For a vector with components (x, y), the magnitude is √(x² + y²). This gives a scalar representing the vector's size.

Component Form of Vectors

Vectors in two dimensions can be expressed in component form as ai + bj, where a and b are the components along the x- and y-axes respectively. Understanding this form allows for straightforward operations like addition, subtraction, and magnitude calculation.