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

Verified step by step guidance

1. Identify the components of vector \( u \) as \( 2i - 5j \).
2. Identify the components of vector \( v \) as \( -3i + 7j \).
3. Subtract the \( i \)-components of \( v \) from \( u \): \( 2 - (-3) \).
4. Subtract the \( j \)-components of \( v \) from \( u \): \( -5 - 7 \).
5. Combine the results to form the new vector: \( (2 - (-3))i + (-5 - 7)j \).

Key Concepts

Vector Subtraction

Vector subtraction involves finding the difference between two vectors by subtracting their corresponding components. For vectors u and v, this means subtracting the i-components and the j-components separately. The result is a new vector that represents the change from vector v to vector u.

Component Form of Vectors

Vectors in two-dimensional space can be expressed in component form as a combination of i (horizontal) and j (vertical) components. For example, the vector u = 2i - 5j has a horizontal component of 2 and a vertical component of -5. Understanding this form is essential for performing operations like addition and subtraction.

Resultant Vector

The resultant vector is the vector that results from the addition or subtraction of two or more vectors. In the case of u - v, the resultant vector will have its own unique components derived from the differences of the corresponding components of u and v. This resultant vector can be interpreted geometrically as the vector that connects the tip of vector v to the tip of vector u.