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Vector addition is one of the fundamental operations in vector algebra. It is performed component-wise. Let's take two vectors, \( \mathbf{b} = (1, 2, 3) \) and \( \mathbf{c} = (3, 2, 1) \). To find their sum, \( \mathbf{b} + \mathbf{c} \), you simply add their corresponding components together. This means adding the x-components, y-components, and z-components separately.

The resulting vector \( \mathbf{b} + \mathbf{c} \) is:

\[ \mathbf{b} + \mathbf{c} = (1 + 3, 2 + 2, 3 + 1) = (4, 4, 4) \]

Vector addition can be visualized as placing the second vector at the tip of the first vector and then drawing the resultant vector from the origin to the tip of the second vector. This is often referred to as the "head-to-tail" method.

Key points:

• Always add corresponding components.
• The result is another vector.
• Visualize it geometrically as connecting vectors from head to tail.

The dot product, also known as the scalar product, is an operation that combines two vectors to produce a scalar quantity. For vectors \( \mathbf{b} = (1, 2, 3) \) and \( \mathbf{c} = (3, 2, 1) \), the dot product is calculated by multiplying each corresponding pair of components and then summing those products. This operation is essential in various applications such as physics, computer graphics, and finding angles between vectors.

Here's how you calculate it:

\[ \mathbf{b} \cdot \mathbf{c} = (1 \cdot 3) + (2 \cdot 2) + (3 \cdot 1) = 3 + 4 + 3 = 10 \]

Key aspects of the dot product:

• Produces a scalar, not a vector.
• Used to determine if vectors are orthogonal (dot product is zero).
• Provides a measure of the vectors' directional alignment.

The cross product is a vector operation that results in another vector that is perpendicular to the plane containing the original vectors. For vectors \( \mathbf{b} = (1, 2, 3) \) and \( \mathbf{c} = (3, 2, 1) \), the cross product is calculated using a determinant. This operation is crucial in physics and engineering for finding torques, areas of parallelograms, and more.

Calculation involves determining the determinant of a matrix composed of both vectors and the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \):

\[ \mathbf{b} \times \mathbf{c} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 2 & 3 \ 3 & 2 & 1 \end{vmatrix} \]
The components are:

• \( \mathbf{i}\)-component: \((2)(1) - (3)(2) = -4\)
• \( \mathbf{j}\)-component: \((1)(1) - (3)(3) = -8\) with sign change to \(+8\)
• \( \mathbf{k}\)-component: \((1)(2) - (2)(3) = -4\)

Thus, the resultant vector is:

\[ \mathbf{b} \times \mathbf{c} = (-4, 8, -4) \]
Features of the cross product:

• Results in a vector perpendicular to the original vectors.
• Magnitude gives the area of the parallelogram formed by the vectors.
• Direction follows the right-hand rule.

Scalar multiplication involves multiplying a vector by a scalar (a single real number). This operation changes the magnitude of the vector but not its direction, unless the scalar is negative, which would reverse the direction. In the given exercise, we perform this operation on vectors \( \mathbf{b} = (1, 2, 3) \) and \( \mathbf{c} = (3, 2, 1) \).

For vector \( \mathbf{b} \), multiply each component by 5:
\[ 5\mathbf{b} = 5 \times (1, 2, 3) = (5, 10, 15) \]
For vector \( \mathbf{c} \), multiply each component by 2:
\[ 2\mathbf{c} = 2 \times (3, 2, 1) = (6, 4, 2) \]
These transformed vectors can further undergo other operations, such as subtraction, resulting in:
\[ 5\mathbf{b} - 2\mathbf{c} = (5 - 6, 10 - 4, 15 - 2) = (-1, 6, 13) \]
Important points about scalar multiplication:

• Changes magnitude, not the direction (unless negative).
• Scaling positive enlarges, scaling negative reverses direction.
• Scalars can be used to resolve vectors into desired lengths.