In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar. 3v - 4w

Key Concepts

Vector Addition and Scalar Multiplication

Vector addition involves combining two or more vectors by adding their corresponding components. Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. Understanding these operations is essential for manipulating vectors in problems like the one presented.

Component Form of Vectors

Vectors can be expressed in component form, typically as 'ai + bj', where 'a' and 'b' are the horizontal and vertical components, respectively. This representation allows for straightforward calculations, such as addition and scalar multiplication, by simply performing arithmetic on the components. In this question, the vectors u, v, and w are given in this form.

Resultant Vector

The resultant vector is the vector that results from the addition of two or more vectors. It represents the cumulative effect of the individual vectors. In the context of the question, calculating 3v - 4w involves finding the resultant vector after scaling vectors v and w and then performing vector subtraction.