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Chapter 7: Problem 12

Under what conditions on the vectors \(v\) and \(w\) in \(R^{3}\) will \(v \times w=0\) ?

Short Answer

Expert verified
The cross products of vectors \(v\) and \(w\) in \(R^{3}\) will be zero under two conditions: either one or both of the vectors are zero vectors, or, if both are non-zero vectors, they are scalar multiples of each other meaning they are parallel.

Step by step solution

01

Cross product of zero vectors

One condition under which \(v \times w = 0\) is if one or both of the vectors \(v\) and \(w\) are zero. A zero vector cross product with any vector or itself is always a zero vector.
02

Cross product of parallel vectors

The other condition under which \(v \times w = 0\) is if \(v\) and \(w\) are not zero vectors but are parallel to each other. In geometrical terms, parallel vectors are those vectors that have the same direction or are in exactly opposite direction. In terms of vector algebra, two vectors are parallel if one is the scalar multiple of the other. Thus if \(v = kw\), for some scalar \(k\), then \(v\) and \(w\) are parallel to each other and their cross product would be zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Zero Vector in Vector Algebra
In vector algebra, the zero vector can be thought of as a vector with no magnitude or direction. It is represented by a vector with all components being zero, such as \( (0, 0, 0) \) in three-dimensional space \( R^{3} \) . When dealing with cross products, one of the properties that is fundamental to understanding is that the cross product of any vector with the zero vector will result in another zero vector. This is because the cross product involves a sine function of the angle between vectors and since the zero vector has no direction, the angle is undefined and the magnitude of cross product defaults to zero.
Parallel Vectors and Their Impact on Cross Products
Two non-zero vectors are parallel if they point in the same, or exactly opposite, direction. A key point in vector algebra is that the cross product of parallel vectors gives us a zero vector. This is because the cross product of two vectors depends on the sine of the angle between them. For parallel vectors, this angle is either \(0\degree\) or \(180\degree\)), and the sine of both these angles is zero, resulting in the product being zero. In mathematical terms, if vector \( v \) is a scalar multiple of vector \( w \) (or vice versa), i.e., \( v = kw \) for some scalar \( k \), then they are parallel and their cross product will be the zero vector.
Vector Algebra: Interplay Between Direction and Magnitude
Vector algebra is a significant branch of mathematics that deals with vectors, which are quantities having both magnitude and direction. Operations in vector algebra include addition, subtraction, and multiplication of vectors, where the result can be a vector (as in vector addition) or a scalar (as in dot product). The cross product, specifically, yields another vector which is perpendicular to the plane formed by the initial two vectors and whose magnitude is proportional to the area of the parallelogram that they span. The direction of the resulting vector is determined by the right-hand rule. Understanding these operations is essential to analyze conditions for when cross products become zero and to solve numerous geometric and physical problems.
The Role of Scalar Multiples in Vector Relationships
In the context of vector algebra, a scalar multiple of a vector is obtained by multiplying every component of the vector by a scalar value. This operation changes the magnitude of the vector but does not affect its direction, unless the scalar is negative, in which case the vector reverses direction. The notion of scalar multiples is crucial when identifying parallel vectors as mentioned earlier. It also can be applied to determine collinearity of points in three-dimensional space and to solving linear equations involving vectors. Scalar multiplication is commutative, associating, and it distributes over vector addition, making it a fundamental operation in the manipulation and understanding of vectors in algebra.

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Most popular questions from this chapter

Show that the composition of orientation-preserving linear maps is orientationpreserving. What about the composition of orientation-reversing linear maps? What if an orientation-preserving linear map is composed with an orientationreversing linear map? Be sure to consider the two possible orders of composing the two maps.

Find a normal vector for the plane $$ \\{r(5,1,7)+s(-3,2,2)+(1,0,-4) \mid r, s \in \mathbb{R}\\} . $$

Fill in the details of this alternative derivation of Cramer's rule for solving \(A \mathbf{x}=\mathbf{b}\). It is based on Stephen Robinson's article "A Short Proof of Cramer's Rule" in Mathematics Magazine, March-April 1979, pages 94-95. a. Let \(X_{j}\) denote the matrix obtained by replacing the jth column of \(I\) with \(\mathrm{x}\). Let \(A_{j}\) be the matrix obtained by replacing the \(j\) th column of \(A\) with \(b\). Verify that \(A X_{j}=A_{j}\). b. Show that \(\operatorname{det} X_{j}=x_{j}\), the \(j\) th component of \(\mathrm{x}\). c. Apply Theorem \(7.7\) to the matrix equation in part a and solve for \(x\).

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