The directions of vectors 𝐀 and 𝐁 are given below for...

The directions of vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ are given below for several cases. For each case, state the direction of $\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}$. (a) $\overrightarrow{\mathbf{A}}$ points east, $\overrightarrow{\mathbf{B}}$ points south. $(b) \overrightarrow{\mathbf{A}}$ points east, $\overrightarrow{\mathbf{B}}$ points straight down. ( $c$ ) $\overrightarrow{\mathbf{A}}$ points straight up, $\overrightarrow{\mathbf{B}}$ points north. (d) $\overrightarrow{\mathbf{A}}$ points straight up, $\overrightarrow{\mathbf{B}}$ points straight down.

The directions of vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ are given below for several cases. For each case, state the direction of $\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}$.
(a) $\overrightarrow{\mathbf{A}}$ points east, $\overrightarrow{\mathbf{B}}$ points south. $(b) \overrightarrow{\mathbf{A}}$ points east, $\overrightarrow{\mathbf{B}}$ points straight down. ( $c$ ) $\overrightarrow{\mathbf{A}}$ points straight up, $\overrightarrow{\mathbf{B}}$ points north.
(d) $\overrightarrow{\mathbf{A}}$ points straight up, $\overrightarrow{\mathbf{B}}$ points straight down.

Key Concepts

Cross Product

The cross product is a mathematical operation on two vectors in three-dimensional space that results in another vector that is perpendicular to the plane containing the original vectors. Its magnitude is equal to the product of the magnitudes of the two vectors times the sine of the angle between them, and it represents the area of the parallelogram formed by the vectors.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the vector resulting from a cross product. By pointing the fingers of the right hand in the direction of the first vector and curling them toward the second vector, the thumb then points in the direction of the cross product, establishing the orientation of the resulting vector.

Parallel and Anti-Parallel Vectors

When two vectors are parallel or anti-parallel, the sine of the angle between them is zero, meaning that the cross product yields a zero vector. This concept highlights that the cross product is non-zero only when the vectors are not aligned, emphasizing the importance of the angle between them in determining the magnitude of the resulting vector.

Transcript

00:01 So in this question we are given various directions of two vectors a and b and we are asked to find the directions of the cross products of these a and b vectors.

00:11 So let's define the directions that we are given.

00:15 So this horizontal plane, let's say this plane that we are talking about, this contains the north direction, the east direction, the south direction and the west direction.

00:30 And perpendicular, these are up direction and down direction.

00:38 So with this let's with these reference directions, let's try to try and solve the problem.

00:46 So over the first part we are told that a is in the east direction and b is in the south direction.

00:54 So if you try to draw a diagram which takes from the reference case, it will give us that this is a pointing to the right, excuse me, and south will be pointing this way in the horizontal plane.

01:08 Sorry, this is b direction south and east.

01:13 So what we will do is we will curl our, we use the right -hand rule, right -hand thumb -brun, and we curl our fingers from a vector to b -vector.

01:20 We are always trying to find direction of a cross b here.

01:25 So we curl our directions, our fingers from the direction of a and b, along the smaller angle between a.

01:31 So there are two angles between a and b.

01:33 This is the smaller angle and this is the larger angle.

01:36 So we take it from the smaller angle to the larger angle.

01:38 So we curl our fingers from a to b and we find there are thumb points downwards if we do that.

01:45 So this is the direction of a cross b, which from our reference case means this answer is down for part a...