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

Key Concepts

Vector Operations

Vector operations involve mathematical manipulations of vectors, such as addition, subtraction, and scalar multiplication. In this case, multiplying vector v by a scalar (5) means scaling its components by that number, which affects both the magnitude and direction of the vector.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar (a real number). This operation results in a new vector whose direction remains the same if the scalar is positive, but its magnitude is scaled by the absolute value of the scalar. For example, multiplying vector v by 5 will increase its length while maintaining its direction.

Component Form of Vectors

Vectors can be expressed in component form, typically as a combination of unit vectors i and j in a two-dimensional space. For instance, vector v = -3i + 7j indicates that it has a horizontal component of -3 and a vertical component of 7. Understanding this representation is crucial for performing operations like scalar multiplication.