Multivariable Calculus · 免费预览

叉积

通过行列式计算两个空间向量的叉积，其大小为平行四边形的面积，其方向由右手定则确定，以及其代数性质。

1. The cross product produces a vector perpendicular to both inputs

The cross product produces a vector perpendicular to both inputs

2. The magnitude is the area of the spanned parallelogram

The magnitude is the area of the spanned parallelogram

3. The right-hand rule fixes the direction

The right-hand rule fixes the direction

4. Areas and normals are the everyday uses

Areas and normals are the everyday uses

5. Computing a cross product

Let $\mathbf{a} = (1, 2, 3)$ and $\mathbf{b} = (2, 1, 4)$ . Which vector equals $\mathbf{a} \times \mathbf{b}$ ?

6. The basis cycle under the right-hand rule

Using the right-hand rule for the standard basis vectors, which of the following equals $\mathbf{k}$ ?

7. Area of a spanned parallelogram

Two edge vectors of a parallelogram are $\mathbf{a} = (2, 0, 0)$ and $\mathbf{b} = (1, 3, 0)$ . What is the area of the parallelogram they span?

8. When the cross product vanishes

For nonzero vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^3$ , under exactly what condition does $\mathbf{a} \times \mathbf{b} = \mathbf{0}$ hold?

9. The third basis product

Using the right-hand-rule cycle for the standard basis, give the vector $\mathbf{k} \times \mathbf{i}$ as a single letter, i, j, or k.

10. A cross product in the xy-plane

With $\mathbf{a} = (2, 1, 0)$ and $\mathbf{b} = (1, 3, 0)$ , give the third component of $\mathbf{a} \times \mathbf{b}$ , as a single number.

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