Question

Find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(\mathbf{z}=2 \mathbf{u}+4 \mathbf{v}-\mathbf{w}\)

Short answer

\(\mathbf{z} = \langle 6,12,-2 \rangle\)

Step-by-step solution

Apply Scalar Multiplication

In the equation given \(\mathbf{z}=2 \mathbf{u}+4 \mathbf{v}-\mathbf{w}\), the first step is to multiply the vector \(\mathbf{u}\) with the scalar 2 and the vector \(\mathbf{v}\) with the scalar 4, resulting in the vectors \(2\mathbf{u}=\langle 2(1),2(2),2(3)\rangle = \langle 2,4,6\rangle\) and \(4\mathbf{v}= \langle 4(2),4(2),4(-1) \rangle = \langle 8,8,-4 \rangle\).

Perform Vector Addition and Subtraction

The next step is to add the vectors obtained from the scalar multiplication and subtract the vector \(\mathbf{w}\) from the sum. This is done component by component i.e., add/subtract the x-components, then the y-components, then the z-components. So \(\mathbf{z}=2 \mathbf{u}+4 \mathbf{v}-\mathbf{w} = \langle 2+8-4,4+8-0,6-4-4 \rangle = \langle 6,12,-2 \rangle\).

Conclude the Result

The operations result in the vector \(\mathbf{z}\) with components 6, 12, and -2. So, \(\mathbf{z} = \langle 6,12,-2 \rangle\).

Key concepts

Scalar Multiplication

Scalar multiplication in vector operations plays a foundational role in manipulating vectors. It involves multiplying a vector by a scalar (a real number), affecting the magnitude of the vector while maintaining its direction unchanged, unless the scalar is negative, in which case the direction is reversed. In our original exercise, \(\mathbf{z}=2 \mathbf{u}+4 \mathbf{v}-\mathbf{w}\),the scalars are 2 and 4. When we multiply \(\mathbf{u}=\langle 1,2,3\rangle\) by 2 and \(\mathbf{v}=\langle 2,2,-1\rangle\) by 4, the vectors scale up accordingly. Imagine stretching or shrinking the arrows representing \(\mathbf{u}\) and \(\mathbf{v}\) without changing the direction they point to.
Key steps for scalar multiplication include multiplying each component of the vector by the scalar. So, 2 \times \langle 1,2,3 \rangle = \langle 2 \times 1, 2 \times 2, 2 \times 3 \rangle = \langle 2,4,6 \rangle,resulting in a new vector with its magnitude altered but its direction the same as \(\mathbf{u}\).

Vector Addition and Subtraction

Vector addition and subtraction let you combine or remove the influence of multiple vectors. In a practical sense, this is how various forces interact in physics or different displacements are combined in navigation. The process is straightforward: you simply add or subtract the corresponding components of each vector to find the result.

For example, to add two vectors, \(\mathbf{a}\) and \(\mathbf{b}\), in three-dimensional space, you would calculate \(\mathbf{a} + \mathbf{b} = \langle a_{1},a_{2},a_{3}\rangle + \langle b_{1},b_{2},b_{3}\rangle = \langle a_{1}+b_{1}, a_{2}+b_{2}, a_{3}+b_{3} \rangle\).

Following this method, the exercise had us combine scaled vectors and then subtract another:
\(2\mathbf{u} + 4\mathbf{v} - \mathbf{w}\).

This operation can be visualized as moving along the direction and length of the first vector, then continuing along the second vector, and finally reversing the direction and moving backwards by the third vector's length.

Vector Components

Vectors are often expressed in terms of their components, which represent their influence along the respective axes in a coordinate system. Each component is a scalar that indicates how far the vector extends along a particular axis. In a three-dimensional space, a vector's components are denoted as \(\langle x,y,z \rangle\), where x, y, and z are the magnitudes in the directions of the x-axis, y-axis, and z-axis, respectively. In the solution for \(\mathbf{z}\), we found the components to be 6, 12, and -2.

Breaking down the vector components

When analyzing the components of \(\mathbf{z}\), we're actually looking at how much \(\mathbf{z}\) is pointing in the direction of each axis. A positive component means the vector points in the positive direction of that axis, whereas a negative component means it points in the negative direction.