In Exercises 39–42, letu = -i + j, v = 3i - 2j, and w = -5j.Find each specified scalar or vector.5u ⋅ (3v - 4w)

Verified step by step guidance

1. First, calculate the scalar multiplication of the vector \( u \) by 5: \( 5u = 5(-i + j) = -5i + 5j \).
2. Next, calculate the vector \( 3v - 4w \). Start by finding \( 3v \): \( 3v = 3(3i - 2j) = 9i - 6j \).
3. Now, calculate \( 4w \): \( 4w = 4(-5j) = -20j \).
4. Subtract \( 4w \) from \( 3v \): \( 3v - 4w = (9i - 6j) - (-20j) = 9i - 6j + 20j = 9i + 14j \).
5. Finally, find the dot product \( 5u \cdot (3v - 4w) \): \((-5i + 5j) \cdot (9i + 14j) = (-5)(9) + (5)(14)\).

Key Concepts

Vector Operations

Understanding vector operations is crucial in this problem. Vectors can be added or subtracted, and scalar multiplication involves multiplying each component of the vector by a scalar. In this case, we need to perform operations on the vectors u, v, and w to find the resultant vector before applying the dot product.

Dot Product

The dot product is a fundamental operation in vector algebra that combines two vectors to produce a scalar. It is calculated by multiplying corresponding components of the vectors and summing the results. The dot product provides insights into the angle between vectors and is essential for determining projections and orthogonality.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, which scales the vector's magnitude without changing its direction. In this exercise, we first need to compute the vector 3v - 4w, and then multiply the resulting vector by the scalar 5u. This concept is key to manipulating vectors before applying the dot product.