Question

\(\underline{\mathrm{x}} \times(\mathrm{y} \cdot \underline{\mathrm{z}})\) is what ? where \(\underline{\mathrm{x}}, \mathrm{y}, \underline{\mathrm{z}} \in \mathrm{R}^{3}\) (a) box product (b) vector (c) scalar (d) none of these

Short answer

The given expression \(\underline{x} \times (y \cdot \underline{z})\) is a vector (option b).

Step-by-step solution

Analyze the given expression

The given expression is: \(\underline{x} \times (y \cdot \underline{z})\). Here, \(\underline{x}\) and \(\underline{z}\) are vectors, and \(y\) is a scalar. We notice that there is a scalar multiplication operation (\(y \cdot \underline{z}\)) followed by a cross product operation (\(\underline{x} \times (...) \)).

Observe the scalar multiplication

First, let's focus on the scalar multiplication: \(y \cdot \underline{z}\). The scalar multiplication of a scalar with a vector results in another vector, with the direction of the original vector unchanged and the magnitude scaled by the scalar amount. Let's call this resulting vector \(\underline{w}\): \(\underline{w} = y \cdot \underline{z}\).

Calculate the cross product

Now we have a new expression to analyze: \(\underline{x} \times \underline{w}\). The cross product of two vectors is another vector, which is orthogonal (perpendicular) to both original vectors. The magnitude of the resulting vector is the product of the magnitudes of the original vectors and the sine of the angle between them.

Determine the type of the expression

The expression \(\underline{x} \times \underline{w}\) is a vector. This is because the cross product of two vectors results in a vector. Therefore, the given expression is a vector. #Conclusion# The given expression \(\underline{x} \times (y \cdot \underline{z})\) is a vector (option b).

Key concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector analysis that involves the multiplication of a vector by a scalar (a real number). This process scales the vector without altering its direction. It is widely used across various fields, including physics, engineering, and computer graphics. Here's how it works:

• Take a scalar, which is simply a number like 2, -1, 0.5, etc.
• Multiply each component of the vector by this scalar.

For example, if you have a vector \( \mathbf{a} = \langle 2, 3, 4 \rangle \) and you multiply it by a scalar \( k = 3 \), the result is \( 3 \cdot \langle 2, 3, 4 \rangle = \langle 6, 9, 12 \rangle \).The new vector \( \langle 6, 9, 12 \rangle \) maintains the same direction as the original vector \( \mathbf{a} \), but its length, or magnitude, is increased by a factor of 3. This principle is useful in scaling vectors to represent forces, velocities, and other quantities where only the magnitude needs adjusting.

Vector Analysis

Vector analysis involves operations and transformations on vectors, which are quantities that have both direction and magnitude. Key operations in vector analysis include addition, subtraction, dot product, and cross product.The cross product, in particular, is used to find a vector orthogonal to two given vectors in 3-dimensional space. Given two vectors \( \mathbf{x} \) and \( \mathbf{z} \), their cross product \( \mathbf{x} \times \mathbf{z} \) results in a vector that is perpendicular to both \( \mathbf{x} \) and \( \mathbf{z} \). The formula for the cross product in component form is:\[ \mathbf{x} \times \mathbf{z} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \ x_1 & x_2 & x_3 \ z_1 & z_2 & z_3 \end{array} \right| \]Where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are unit vectors and \( x_1, x_2, x_3 \) and \( z_1, z_2, z_3 \) are the components of vectors \( \mathbf{x} \) and \( \mathbf{z} \).In practical terms, vector analysis helps in understanding the spatial relationships between physical phenomena, like determining the direction of torque or the motion of particles in a magnetic field.

Orthogonal Vectors

Orthogonal vectors are vectors that are perpendicular to each other. A pair of orthogonal vectors has a dot product of zero, which is a useful property in various mathematical and engineering applications.Consider two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \). They are orthogonal if:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 = 0 \]This perpendicular relation is especially important in cross product calculation, as the result of a cross product is always orthogonal to the original vectors considered.Applications of orthogonal vectors are numerous:

• In computer graphics to determine surface normals for lighting calculations.
• In physics to represent independent forces.
• In engineering for resolving components into perpendicular directions for manageable calculations.

Understanding orthogonal vectors simplifies many problems, allowing for clear, component-based solutions.