cross product vector problem

Question

I don't understand the book's explanation.

1) So, c = a times b times sin angle. Is this the formula for solving that kind of problem? Yet the last problem I did about this had cos instead of sin -

what's the difference?

2) How is it determined that $90$ is the angle? It fails to explain that.

3) Where does the line $"250 - 90 = 160"$ come from? All I see in the problem are $250$, $18$, and $12$. There's no $90$. Where is the book getting $90$ from?

4) How is it determined that the $160$ degrees is in "positive x axis"? What indicates that?

Answer 1

1) The first image you posted discusses the cross product, the second the scalar (aka "dot") product. This explains the cosine/sine confusion.

2) The problem specifies that $\vec{a}$ is in the $xy$-plane and $\vec{b}$ is parallel to the $z$-axis, so they must be perpendicular (i.e. make a 90-degree angle).

3) The formula they're using is $$\text{angle from $x$-axis to $\vec{a}$} = \text{angle from $x$-axis to $\vec{c}$} + \text{angle from $\vec{c}$ to $\vec{a}$} .$$ We're guaranteed that the cross-product $\vec{c} = \vec{a} \times \vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$, so $\text{angle from $\vec{c}$ to $\vec{a}$} = 90^\circ$ (well, it would be $-90^\circ$ instead if $\vec{b}$ pointed along $-\vec{z}$ instead of $+\vec{z}$; this is why drawing diagrams is helpful).

4) Angles in the $xy$-plane are measured from the positive $x$-axis by convention. There's nothing more to it.

Answer 2

1. Known fact to learn: $||a \times b|| = ||a|| ||b|| sin \theta$ where $\theta$ is the angle between $a$ and $b$.

   Note 1-1: $a$ and $b$ are vectors; otherwise this would not make sense

Note 1-2: Always read $a \times b$ as $a$ cross $b$. It is not real number multiplication and you want to separate the ideas in your mind.

Note 1-3: $||x||$ is the magnitude or length of the vector $x$. Always say this as you read or else the rest makes no sense.

2. Fact to learn: The cross product $a \times b$ is always orthogonal (at right angles in 3D) to $a$ and $b$ so it is at 90 degrees to the plane containing them. The vector $b$ is given as being on the positive z axis so it points straight up from the table surface. $a \times b$ is 90 degrees from b so it is on the table surface or the xy plane, and it is 90 degrees from a and that is where the 90 comes from.

3. Vector $a$ was given in the problem as being at an angle of 250 degrees (always say degrees)measured from the positive x-axis. This is the standard way to measure angles. Our cross product is 90 degrees away from that by the way cross products are made and the text shows the direction so we subtract 250 degrees - 90 degrees = 160 degrees measured. Be careful to relate degrees to degrees and magnitudes/lengths to each other.

4, Angles are usually measured from the positive x axis as a standard so we all understand each other,