Area identities and a cross-product degeneracy

\mathbf c=\mathbf a+\mathbf b

Total marks: 5

Solution

(i)

Distance from a line

State the shortest distance from $B$ to the line through $O$ and $A$ .

Explore this idea

Hint

$\frac{|\mathbf b\times\mathbf a|}{|\mathbf a|}$ .

Answer · (i)

|b\times a|/|a|

Explore the idea

Distance from B to the line OA

Switch parts of the scene on one at a time and read how each is found. Drag to turn the picture.

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(ii)

Area identity

With $\mathbf c=\mathbf a+\mathbf b$ , state the geometrical meaning of $|\mathbf b\times\mathbf c|$ and show it equals $|\mathbf a\times\mathbf b|$ .

Explore this idea

Hint

$|\mathbf b\times\mathbf c|$ is the area of the parallelogram with sides $OB$ and $OC$ .
Expand using $\mathbf b\times\mathbf b=\mathbf 0$ .

Answer · (ii)

OB,OC

Explore the idea

Slide the side, keep the area

Open the interactive view to explore this geometry in 3D.

(iii)

Degenerate cross product

Given $\mathbf a\times2\mathbf b=\mathbf d\times3\mathbf a$ , find a linear relationship between $\mathbf a$ , $\mathbf b$ , $\mathbf d$ .

Hint

Move everything to one side and factor out $\mathbf a\times$ .
$\mathbf a\times\mathbf w=\mathbf 0$ means $\mathbf a\parallel\mathbf w$ .

Answer · (iii)

2b+3d=ta

Area identities and a cross-product degeneracy

Solution

Distance from a line

Formula

Distance from B to the line OA

Area identity

Meaning

Expand

Magnitude

Slide the side, keep the area

Degenerate cross product

Rearrange

Parallel