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 \(u\) points east and \(v\) points south, then \(u \times v\) 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 \(\mathbf{v}=\mathbf{0}\) e. Law of Cancellation? If \(\mathbf{u} \times \mathbf{v}=\mathbf{u} \times \mathbf{w},\) then \(\mathbf{v}=\mathbf{w}\)

Short answer

Question: Analyze the following statements and determine if they are true or false: a. Cross product of two nonzero vectors will always be nonzero. b. The magnitude of the cross product of two vectors cannot be greater than the product of magnitudes of both input vectors. c. If a vector is pointing east and another is pointing south, their cross product will always be pointing west. d. If the cross product and dot product of two vectors are both zero, one of the vectors must be the zero vector. e. Cancellation law holds for vector cross products. Answer: a. False b. True c. False d. True e. False

Step-by-step solution

a. Cross product of two nonzero vectors

The cross product of two nonzero vectors will be a nonzero vector if the vectors are linearly independent (not scalar multiples of each other). If the vectors are scalar multiples of each other, the cross product will be a zero vector since their plane is degenerate. Consider vectors \(\mathbf{u} = (1,0,0)\) and \(\mathbf{v} = (2,0,0)\). Their cross product \(\mathbf{u} \times \mathbf{v}= (0,0,0)\) is a zero vector. Therefore, the statement is false.

b. Magnitude relationship

The magnitude of the cross product \(|\mathbf{u} \times \mathbf{v}|\) is given by \(|\mathbf{u}||\mathbf{v}|\sin{\theta}\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). The sine function is bounded between \(0\) and \(1\), so \(|\mathbf{u} \times \mathbf{v}| \le |\mathbf{u}||\mathbf{v}|\). Therefore, the magnitude of cross product cannot be greater than the product of magnitudes of both input vectors. However, it can be equal if the angle between them is \(90^\circ\). So, the statement is true.

c. Direction of cross product

Let \(\mathbf{u}\) be a vector pointing east and \(\mathbf{v}\) be a vector pointing south. Their cross product \(\mathbf{u} \times \mathbf{v}\) will be orthogonal to the plane formed by \(\mathbf{u}\) and \(\mathbf{v}\), following the right hand rule. According to this rule, if your fingers point from \(\mathbf{u}\) to \(\mathbf{v}\), your thumb will point in the direction of \(\mathbf{u} \times \mathbf{v}\). Since the thumb will point upward in this case, the statement is false. Instead, \(\mathbf{u} \times \mathbf{v}\) will point upward, not west.

d. Cross product and dot product conditions

If \(\mathbf{u} \times \mathbf{v} = \mathbf{0}\), then either \(\mathbf{u}\) and \(\mathbf{v}\) are parallel, or at least one of them is the zero vector. If \(\mathbf{u} \cdot \mathbf{v} = 0\), then \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal (perpendicular) to each other, or at least one of them is the zero vector. If you have both conditions satisfied, then it means that one of the vectors must be \(\mathbf{0}\). So the statement is true.

e. Cancellation law

If \(\mathbf{u} \times \mathbf{v} = \mathbf{u} \times \mathbf{w}\), it does not imply that \(\mathbf{v} = \mathbf{w}\). Consider vectors \(\mathbf{u} = (1,0,0)\), \(\mathbf{v} = (0,1,0)\), and \(\mathbf{w} = (0,1,2)\). The cross product \(\mathbf{u} \times \mathbf{v} = (0,0,1)\) and \(\mathbf{u} \times \mathbf{w} = (0,0,1)\), but \(\mathbf{v} \ne \mathbf{w}\). So the statement is false, and the cancellation law does not hold for vector cross products.

Key concepts

Vector Magnitude

In the context of vectors, the term 'magnitude' refers to the size or length of a vector. Imagine you're drawing an arrow; the length of this arrow represents the vector magnitude. Mathematically, for a vector \( \mathbf{v} = (x,y,z) \), the magnitude is calculated using the formula \( |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} \). Understanding magnitude is crucial when analyzing the cross product of vectors, as it's a measure of the vector's 'strength' in a particular direction.
This concept directly relates to the exploration of the magnitude of a cross product, which, as specified in the exercise, is influenced by the sine of the angle between those vectors and the magnitudes of the original vectors involved.

Right Hand Rule

The right hand rule is a handy mnemonic for understanding the direction of the cross product of two vectors. Here's how it works: point your index finger in the direction of the first vector (\(\mathbf{u}\)), then point your middle finger in the direction of the second vector (\(\mathbf{v}\)). Your thumb now points in the direction of the cross product (\(\mathbf{u} \times \mathbf{v}\)).
It's crucial when determining whether the resultant vector points up or down, east or west, or any other direction in 3D space. This exercise specifically calls it into question for vectors pointing in cardinal directions, leading to the conclusion that when vector \(\mathbf{u}\) points east and vector \(\mathbf{v}\) points south, their cross product points upwards, not west.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. This operation is defined algebraically by \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \), and geometrically as \( |\mathbf{u}| |\mathbf{v}| \cos \theta \), where \(\theta\) is the angle between the vectors. It's useful for determining the angle between vectors and for checking if two vectors are orthogonal since the dot product of orthogonal vectors is zero.
In the exercise, understanding the dot product is essential for proving that vectors are orthogonal, which in turn, affects the interpretation of their cross product.

Orthogonal Vectors

Vectors are orthogonal if they meet at a right angle (90 degrees). The dot product comes into play here: if the dot product of two vectors is zero, this indicates that they are orthogonal. Orthogonality is crucial for the cross product because if vectors are orthogonal, the magnitude of their cross product will be maximized, directly proportional to the product of the magnitudes of the original vectors.
As seen in the provided solutions, identification of orthogonal vectors (such as by the dot product being zero) aids in determining the conditions when cross or dot product results in the zero vector. In our exercise scenario, if \(\mathbf{u} \times \mathbf{v}=\mathbf{0}\) and \(\mathbf{u} \cdot \mathbf{v}=0\), we conclude that the vectors are both parallel (or one is zero) and orthogonal, which can only happen if one of the vectors is indeed the zero vector.

Linear Independence

Linear independence is a property of vectors where no vector in the set can be written as a linear combination of the others. For two vectors, this simply means they do not fall along the same line and are not multiples of each other. It's a foundational concept for cross products because the cross product of linearly independent vectors in three-dimensional space results in a nonzero vector that is orthogonal to the plane containing the original vectors.
In the solutions we've seen, linear independence determines the truth of the statement about the cross product of two nonzero vectors resulting in a nonzero vector. Linear independence assures us that the cross product provides meaningful geometric information about the vectors' orientation in space.