Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The cross product of two nonzero vectors is a nonzero vector. b. \(|\mathbf{u} \times \mathbf{v}|\) is less than both \(|\mathbf{u}|\) and \(|\mathbf{v}|\) c. If \(\mathbf{u}\) points east and \(\mathbf{v}\) points south, then \(\mathbf{u} \times \mathbf{v}\) points west. d. If \(\mathbf{u} \times \mathbf{v}=\mathbf{0}\) and \(\mathbf{u} \cdot \mathbf{v}=0,\) then either \(\mathbf{u}=\mathbf{0}\) or \(\mathbf{v}=\mathbf{0}\) (or both). e. Law of Cancellation? If \(\mathbf{u} \times \mathbf{v}=\mathbf{u} \times \mathbf{w},\) then \(\mathbf{v}=\mathbf{w}\)

Short answer

Short answer: a) False. The cross product of two parallel nonzero vectors equals the zero vector. For example, \(\mathbf{u} = (1, 0, 0)\) and \(\mathbf{v} = (2, 0, 0)\) gives \(\mathbf{u} \times \mathbf{v} = (0, 0, 0)\). b) True. The magnitude of the cross product is given by \(|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin(\theta),\) and since the sine function ranges between -1 and 1, the magnitude of the cross product cannot be greater than the product of the magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\). c) False. If \(\mathbf{u} = (1, 0, 0)\) and \(\mathbf{v} = (0, -1, 0)\), then \(\mathbf{u} \times \mathbf{v} = (0, 0, -1)\), which points downward, not west. d) False. Orthogonal nonzero vectors \(\mathbf{u}=(1, 0, 0)\) and \(\mathbf{v}=(0, 1, 0)\) satisfy \(\mathbf{u}\cdot\mathbf{v}=0\) and \(\mathbf{u}\times\mathbf{v}=(0, 0, 1)\), but neither \(\mathbf{u}\) nor \(\mathbf{v}\) is a zero vector. e) False. For \(\mathbf{u}=(0, 0, 1)\), \(\mathbf{v}=(1, 0, 0)\), and \(\mathbf{w}=(0,1,0)\), we have \(\mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w} = (0, -1, 0)\), but \(\mathbf{v}\neq\mathbf{w}\).

Step-by-step solution

Statement a

The cross product of two nonzero vectors is a nonzero vector. Our task is to determine if this statement is true or false and provide a counterexample if it's false. This statement is not always true. The cross product of two nonzero vectors can be zero if and only if they are parallel. Here is a counterexample: consider vectors \(\mathbf{u} = (1, 0, 0)\) and \(\mathbf{v} = (2, 0, 0)\). The cross product \(\mathbf{u} \times \mathbf{v}\) equals \((0, 0, 0)\) which is a zero vector.

Statement b

\(|\mathbf{u} \times \mathbf{v}|\) is less than both \(|\mathbf{u}|\) and \(|\mathbf{v}|\). Our task is to determine if this statement is true or false and provide an explanation if it's true or a counterexample if it's false. This statement is true. The magnitude of the cross product is given by the formula: \(|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\sin(\theta),\) where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}.\) Since the sine function ranges between -1 and 1, the magnitude of the cross product cannot be greater than the product of the magnitudes of \(\mathbf{u}\) and \(\mathbf{v}.\)

Statement c

If \(\mathbf{u}\) points east and \(\mathbf{v}\) points south, then \(\mathbf{u} \times \mathbf{v}\) points west. Our task is to determine if this statement is true or false and provide an explanation if it's true or a counterexample if it's false. This statement is false. For illustrative purposes, let \(\mathbf{u} = (1, 0, 0)\) and \(\mathbf{v}= (0, -1, 0)\). The cross product \(\mathbf{u} \times \mathbf{v} = (0, 0, -1)\), which points in the downward direction (opposite of the upward direction), not west.

Statement d

If \(\mathbf{u}\times\mathbf{v}=\mathbf{0}\) and \(\mathbf{u}\cdot\mathbf{v}=0,\) then either \(\mathbf{u}=\mathbf{0}\) or \(\mathbf{v}=\mathbf{0}\) (or both). Our task is to determine if this statement is true or false and provide an explanation if it's true or a counterexample if it's false. This statement is false. Consider two orthogonal (perpendicular) nonzero vectors, \(\mathbf{u}=(1, 0, 0)\) and \(\mathbf{v}=(0, 1, 0)\). Clearly, \(\mathbf{u}\cdot\mathbf{v}=0\), and from our previous analysis \(\mathbf{u}\times\mathbf{v}=(0, 0, 1)\). We can see that neither \(\mathbf{u}=\mathbf{0}\) nor \(\mathbf{v}=\mathbf{0}\).

Statement e

Law of Cancellation - If \(\mathbf{u} \times \mathbf{v}=\mathbf{u} \times \mathbf{w},\) then \(\mathbf{v}=\mathbf{w}\). Our task is to determine if this statement is true or false and provide an explanation if it's true or a counterexample if it's false. This statement is false. Consider \(\mathbf{u}=(0, 0, 1)\), \(\mathbf{v}=(1, 0, 0)\), and \(\mathbf{w}=(0,1,0)\). We have \(\mathbf{u} \times \mathbf{v} = (0, -1, 0)\) and \(\mathbf{u} \times \mathbf{w} = (0, -1, 0)\), but \(\mathbf{v}\neq\mathbf{w}\).

Key concepts

Vector Magnitude

The magnitude of a vector refers to how long or large the vector is, and it is always a non-negative number. Think of vector magnitude as the length of the arrow representing the vector.

• Mathematically, for a vector \( \mathbf{u} = (u_1, u_2, u_3) \), the magnitude \( |\mathbf{u}| \) is calculated using the formula: \( |\mathbf{u}| = \sqrt{u_1^2 + u_2^2 + u_3^2} \).
• This can also be understood through the concept of the distance formula in 3D space.
• In cross product calculations, understanding vector magnitude is crucial as it relates to how "strong" or significant a vector is in space.

Keep in mind that the magnitude of the cross product \(|\mathbf{u} \times \mathbf{v}|\) is also dependent on the sine of the angle between vectors \(\mathbf{u}\) and \(\mathbf{v}\). This means it results in maximum value when the vectors are perpendicular and zero if they are parallel.

Orthogonal Vectors

Orthogonal Vectors are vectors that meet at a right angle or 90 degrees. When two vectors are orthogonal, they are said to be perpendicular.

• Mathematically, two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal if and only if their dot product is zero: \( \mathbf{u} \cdot \mathbf{v} = 0 \).
• If two vectors are orthogonal, their cross product gives a vector whose magnitude is maximized.
• The cross product \( \mathbf{u} \times \mathbf{v} \) results in a third vector that is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \).

Orthogonality is particularly useful in physics and engineering, where it frequently represents independence of components.

Vector Calculus

Vector Calculus is a branch of mathematics that deals with vector fields and differentiating or integrating vectors.

• It is used to analyze and solve physical problems, particularly those with directions, such as electromagnetism or fluid dynamics.
• Cross products form a key aspect of vector calculus, allowing us to compute areas of parallelograms formed by vectors or to find orthogonal vectors.
• Understanding integration and differentiation of vectors helps in calculating quantities like work done by a force field or the flow across a surface.

In vector calculus, the interaction of vectors and their directional properties are explored, often using operations like the dot product and cross product to analyze these behaviors.

Vector Direction

Direction is an essential aspect of vectors as it describes where the vector is pointing.

• Vectors not only have magnitude but also a specified direction. This direction is usually represented by the angle it forms with a reference axis.
• The direction of a cross product \( \mathbf{u} \times \mathbf{v} \) is perpendicular to the plane formed by \( \mathbf{u} \) and \( \mathbf{v} \), following the right-hand rule.
• The right-hand rule is a way to determine the direction of the third vector. By pointing your fingers in the direction of \( \mathbf{u} \) and curling them towards \( \mathbf{v} \), your thumb points in the direction of \( \mathbf{u} \times \mathbf{v} \).

Understanding vector direction is key in navigation, physics simulations, and computer graphics, where knowing the rotation and orientation of objects is essential.