Textbook Question
In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.
3w + 2v
742
views
8. Vectors
Geometric Vectors
2:26 minutesProblem 37
Textbook Question
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 guidance1
Identify the vectors given: \( \mathbf{u} = 2\mathbf{i} - 5\mathbf{j} \), \( \mathbf{w} = -\mathbf{i} - 6\mathbf{j} \).
Calculate the vector difference \( \mathbf{w} - \mathbf{u} \) by subtracting the components of \( \mathbf{u} \) from \( \mathbf{w} \):
\[ \mathbf{w} - \mathbf{u} = (-1 - 2)\mathbf{i} + (-6 - (-5))\mathbf{j} \]
Simplify the components to get the resulting vector:
\[ \mathbf{w} - \mathbf{u} = (-3)\mathbf{i} + (-1)\mathbf{j} \]
Find the magnitude (or norm) of the vector \( \mathbf{w} - \mathbf{u} \) using the formula for the length of a vector in 2D:
\[ ||\mathbf{w} - \mathbf{u}|| = \sqrt{(-3)^2 + (-1)^2} \]
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
05:29
Adding Vectors Geometrically
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.
Recommended video:
04:44Finding Magnitude of a Vector
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.
Recommended video:
03:55Position Vectors & Component Form
Related Videos
Related Practice