Question

Which of the following do not make sense? (a) \(\mathbf{u} \cdot(\mathbf{v} \cdot \mathbf{w})\) (b) \((\mathbf{u} \cdot \mathbf{w})+\mathbf{w}\) \((\mathrm{c})\|\mathbf{u}\|(\mathbf{v} \cdot \mathbf{w})\) (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\)

Short answer

Expressions (a) and (b) do not make sense.

Step-by-step solution

Identify Vector and Scalar Operations

Review the operations involved in each expression. Recall that the dot product \( \mathbf{u} \cdot \mathbf{v} \) results in a scalar, not a vector. In this exercise, look for operations that imply an incorrect sequence of operations between vectors and scalars.

Analyze Option (a)

Option (a) is \( \mathbf{u} \cdot(\mathbf{v} \cdot \mathbf{w}) \). Here, \( \mathbf{v} \cdot \mathbf{w} \) computes a scalar, but \( \mathbf{u} \cdot s \) (where \( s \) is a scalar) is not defined, as the dot product requires two vectors. Hence, this expression does not make sense.

Analyze Option (b)

Option (b) is \( (\mathbf{u} \cdot \mathbf{w}) + \mathbf{w} \). \( \mathbf{u} \cdot \mathbf{w} \) results in a scalar, and adding a scalar to a vector (\( \mathbf{w} \)) is not defined in vector arithmetic. Hence, this expression does not make sense.

Analyze Option (c)

Option (c) is \( \|\mathbf{u}\|(\mathbf{v} \cdot \mathbf{w}) \). The magnitude \( \|\mathbf{u}\| \) is a scalar, and \( \mathbf{v} \cdot \mathbf{w} \) is also a scalar, so this results in a multiplication of two scalars, which makes sense.

Analyze Option (d)

Option (d) is \( (\mathbf{u} \cdot \mathbf{v}) \mathbf{w} \). The dot product \( \mathbf{u} \cdot \mathbf{v} \) is a scalar, and it can be multiplied by the vector \( \mathbf{w} \), resulting in a vector. Hence, this expression makes sense.

Key concepts

Dot Product

In vector calculus, the dot product is a powerful tool used to simplify computations and express certain mathematical phenomena. The dot product of two vectors, say \( \mathbf{u} \) and \( \mathbf{v} \), is a scalar value and is defined as: \[\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \]where \( u_1, u_2, u_3 \) and \( v_1, v_2, v_3 \) are the components of the vectors \( \mathbf{u} \) and \( \mathbf{v} \) respectively. This results in a single number, not a vector. So, it is important to remember that you cannot perform vector operations on it, such as taking the dot product of a scalar and a vector, or adding it directly to a vector.

• **Common mistake**: Trying to compute the dot product between a scalar and a vector, or vice versa.
• **Key takeaway**: Only apply the dot product to pairs of vectors to produce useful scalar results.

Vector Arithmetic

Vector arithmetic encompasses various operations like addition, subtraction, and scalar multiplication on vectors. In vector addition, two vectors are combined to form a third vector. This operation is performed component-wise, meaning: \[\mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3). \]Similarly, subtraction follows the same component-wise approach. One must be careful when working with vectors and scalars; they cannot be mixed directly. For instance, trying to add a scalar directly to a vector, as shown in option (b) of the exercise, is not allowed. The vector addition rule applies strictly to vectors, meaning both quantities in the operation must have vector properties.

• **Important Rule**: Only vectors can be added or subtracted from each other. Scalars require a different approach, like scaling vectors.
• **Always Remember**: Treat vectors and scalars according to their unique rules to ensure proper calculations.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar, leading to a change in the magnitude of the vector but not its direction (except when the scalar is negative, which reverses the direction). For example, when a vector \( \mathbf{v} \) is multiplied by a scalar \( \alpha \), the result is:\[\alpha \mathbf{v} = (\alpha v_1, \alpha v_2, \alpha v_3). \]In scalar multiplication, each component of the vector is multiplied by the scalar. This operation is quite different from the dot product, where two vectors come together to produce a scalar result. a scalar-scalar multiplication, which appears in option (c) in the original exercise, is straightforward; the result is simply another scalar.

• **Key Notion**: In scalar multiplication, the scalar affects the vector uniformly, scaling the entire vector up or down.
• **Easy Mistake**: Confusing scalar multiplication with vector addition or other vector-based operations.

Mathematical Expressions

Mathematical expressions in vector calculus must adhere to strict operational rules that dictate how and when different operations can be applied. Each element in an expression, whether it's a scalar or a vector, plays a specific role dictated by mathematical rules. Operations like dot products must lead to meaningful mathematical outcomes. For example, expressions must conform to structure and type requirements. When mixing various operations like those found in expressions in vector calculus, it is crucial to:

• **Ensure Validity**: Confirm that operations are logical; e.g., the dot product requires two vectors, aiming to produce a scalar, which can then only be added, multiplied, or otherwise manipulated within the boundaries of mathematical laws.
• **Type Consistency**: Ensure the results of different operations are consistent with what's expected in further mathematical manipulations (e.g., scalar vs. vector).
• **Simplify Correctly**: Be sure to simplify expressions according to valid mathematical and algebraic rules, avoiding missteps, like improperly simplifying vector expressions.

These concepts collectively underline the importance of logical consistency and rule adherence in vector calculus.