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

Verified step by step guidance

1. Identify the vector w, which is given as \(-\mathbf{i} - 6\mathbf{j}\).
2. To find \(-4\mathbf{w}\), multiply each component of the vector \(\mathbf{w}\) by \(-4\).
3. Multiply the \(\mathbf{i}\) component: \(-4 \times (-1)\).
4. Multiply the \(\mathbf{j}\) component: \(-4 \times (-6)\).
5. Combine the results to express the vector \(-4\mathbf{w}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\).

Key Concepts

Vector Operations

Vector operations include addition, subtraction, and scalar multiplication. In this context, scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude while maintaining its direction. For example, multiplying a vector by -4 will reverse its direction and increase its length by a factor of 4.

Vector Representation

Vectors in two-dimensional space can be represented in the form of 'ai + bj', where 'a' and 'b' are the components along the x-axis and y-axis, respectively. In this question, the vectors u, v, and w are expressed in this format, allowing for straightforward manipulation and calculation of their properties.

Scalar Multiplication of Vectors

Scalar multiplication of a vector involves multiplying each component of the vector by the scalar. For instance, if we have a vector w = -i - 6j and we multiply it by -4, we calculate -4 * (-1) for the i component and -4 * (-6) for the j component, resulting in a new vector that reflects the scaling effect of the scalar.