If you're familiar with determinants, show that the cross...

If you're familiar with determinants, show that the cross product can be written as a determinant: $$\vec{A} \times \vec{B}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\A_{x} & A_{y} & A_{z} \\B_{x} & B_{y} & B_{z}\end{array}\right|$$ (Hint: See the preceding problem.)

Key Concepts

Cross Product

The cross product is an operation on two three-dimensional vectors that results in a new vector perpendicular to the plane containing the original vectors. Its magnitude corresponds to the area of the parallelogram spanned by the vectors, and its direction is determined by the right-hand rule. Expressing the cross product as a determinant organizes the computation into the evaluation of minors, facilitating a systematic method to derive the components of the resulting vector.

Determinants

A determinant is a scalar value computed from a square matrix that summarizes certain properties of the matrix, such as invertibility and volume scaling in linear transformations. In the context of the cross product, the determinant provides an organized way to combine the components of two vectors using the properties of matrix expansion, specifically by calculating the signed volume of the parallelepiped defined by basis vectors and the two given vectors.

Transcript

00:01 Okay, so we want to show that the cross product can be written as the determinant of this matrix.

00:05 I always remember it by, i remember the determinant way first and not the other way.

00:14 So anyway, i've been doing this for all the videos, but i guess i can kind of show it.

00:21 So a cross b is, we want to show us to determine of this matrix.

00:28 So we'll put a little question mark.

00:31 So i -hat, j -hat, k -hat, and then a -x, a -y, a -z, b -x, b -y, b -z, and basically to get the determinant of this matrix, i forgot what it's called.

00:57 I want to say expansion by minors, i'm not totally sure.

01:00 But basically you imagine like sort of, i'm just going to draw this out and then like undo it.

01:07 You imagine like crossing these out and then sort of circling this.

01:13 And you basically first find the determinant of this little sub matrix, which is just ay, b, z minus az b, y, and then multiplied by i -hat.

01:24 And then what we'll do next is we'll do negative.

01:31 We'll do the same thing.

01:32 We'll cross out these two and then sort of have this be the thing that we multiply the matrix by.

01:39 So then the next matrix to find the determinant of is this one with ax, az, bx, bx, b, z.

01:45 Like we just kind of pretend it's all squished together in one, two -by -two matrix.

01:50 And then so that's a -x times b -z minus a -z times b -x, just by the rules of the determinants of two -by -two matrices...

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