# Enrichment, Extension, and Application

1 point

## Exercises

1. Two $$3$$-dimensional vectors, $$\vec{u}$$ and $$\vec{v}$$, are placed tail to tail. Describe, geometrically, all three dimensional vectors $$\vec{w}$$ that, when placed tail to tail with $$\vec{u}$$ and $$\vec{v}$$, form the same angle with both vectors simultaneously.
2. (Note: These results may be used in later problems) Prove the following statements:
Statement 1: If point $$P$$ divides the segment $$AB$$ in the ratio $$a : b$$, then for any point $$O$$, we have $\overrightarrow{OP} = \dfrac{b}{a + b}\overrightarrow{OA} + \dfrac{a}{a + b}\overrightarrow{OB}$
Statement 2: If, for a point $$O$$, we have $$\overrightarrow{OP} = b\overrightarrow{OA} + a\overrightarrow{OB}$$ with $$a + b = 1$$, then $$A$$, $$P$$, and $$B$$ are collinear and $$P$$ divides the segment $$AB$$ in the ratio $$a : b$$.
3. In a quadrilateral $$ABCD$$, $$T$$ is the midpoint of the side $$AB$$, $$U$$ is the midpoint of the side $$CD$$, $$L$$ is the midpoint of the diagonal $$AC$$, and $$M$$ is the midpoint of the diagonal $$BD$$. Denote $$\overrightarrow{AB} = \vec{a}$$, $$\overrightarrow{BC} = \vec{b}$$, and $$\overrightarrow{CD} = \vec{c}$$.
1. Show that $$\overrightarrow{AD} + \overrightarrow{BC} = 2\overrightarrow{TU}$$.
2. Show that $$\overrightarrow{AD} + \overrightarrow{CB} = 2\overrightarrow{LM}$$.
3. If $$ABCD$$ is a trapezoid, with parallel bases $$BC = 5$$ and $$AD = 9$$, find
1. the length of the line segment connecting the midpoints of the non-parallel sides; and
2. the length of the line segment connecting the midpoints of the diagonals.
4. In $$\triangle{ABC}$$, $$AM$$, $$BN$$, and $$CP$$ are medians. Prove that $$\overrightarrow{AM} + \overrightarrow{BN} + \overrightarrow{CP} = \vec{0}$$.
5. In $$\triangle{ABC}$$, the altitudes from $$B$$ and $$C$$ meet at point $$H$$. Denote $$\overrightarrow{AB} = \vec{b}$$, $$\overrightarrow{AC} = \vec{c}$$, and $$\overrightarrow{AH} = \vec{h}$$.
1. Express $$\overrightarrow{BH}$$ and $$\overrightarrow{CH}$$ in terms of $$\vec{b}$$, $$\vec{c}$$, and $$\vec{h}$$.
1. Use the dot product to express the fact that $$\overrightarrow{BH}$$ and $$\overrightarrow{CH}$$ are altitudes.
2. Use the equations in part i) to show that $$AH$$ is also an altitude, and hence that the altitudes in a triangle are concurrent.
6. Definition: A vector $$\vec{u}$$ is said to be a linear combination of some (other) vectors $$\vec{a}, \vec{b}, \dots$$ if $$\vec{u}$$ can be written as $\vec{u} = k_1\vec{a} + k_2\vec{b} + \cdots$ where $$k_1, k_2, \dots$$ are constants.

A non-zero vector situated on a line is said to be a base for the line, because any other vector on the line can be expressed uniquely in terms of the given vector. In higher dimensions, two non-collinear vectors in a plane form a base for the plane, because any vector in the plane can be expressed uniquely as a linear combination of those two vectors; similarly, three non-coplanar vectors form a base for the (three-dimensional) space.

1. In the diagram, $$ABCDEFGH$$ is a regular octagon (recall: a regular octagon has all sides equal length and all angles equal measure), as shown below. Let $$\overrightarrow{AB} = \vec{a}$$ and $$\overrightarrow{AH} = \vec{b}$$. Since these vectors are not collinear, they form a basis for the plane, and all other vectors in the plane can be expressed as linear combinations of $$\vec{a}$$ and $$\vec{b}$$. Express the vectors along the other sides of the octagon ( $$\overrightarrow{BC}, \overrightarrow{CD}, \overrightarrow{DE}, \dots$$) as a sum of scalar multiples of $$\vec{a}$$ and $$\vec{b}$$.

2. Consider the cube $$ABCDEFGO$$ shown. The vectors $$\vec{i} = \overrightarrow{OE}, \vec{j} = \overrightarrow{OG}$$, and $$\vec{k} = \overrightarrow{OD}$$ form a basis for the space as they are non-coplanar. Therefore, all vectors (in three-dimensional space) can be expressed uniquely as linear combinations of $$\vec{i}$$, $$\vec{j}$$, and $$\vec{k}$$.

1. Express the vectors $$\vec{a} = \overrightarrow{OA}$$, $$\vec{b} = \overrightarrow{OB}$$, and $$\vec{c} = \overrightarrow{OC}$$ as linear combinations of $$\vec{i}$$, $$\vec{j}$$, and $$\vec{k}$$.
2. Observe that no pair of the vectors $$\vec{a}, \vec{b}, \vec{c}$$ are collinear. Prove that they are not coplanar by showing that $$\vec{b}$$ cannot be written as a linear combination of $$\vec{a}$$ and $$\vec{c}$$.
3. As a result of part b), $$\vec{a}, \vec{b}, \vec{c}$$ also form a basis for the space. Express $$\vec{i}$$, $$\vec{j}$$, and $$\vec{k}$$ as linear combinations of $$\vec{a}$$, $$\vec{b}$$, and  $$\vec{c}$$.
4. Express the vector $$\overrightarrow{AG}$$ as a linear combination of $$\vec{i}$$, $$\vec{j}$$, and $$\vec{k}$$
5. Express the vector $$\overrightarrow{AG}$$ as a linear combination of $$\vec{a}$$, $$\vec{b}$$, and $$\vec{c}$$
7. In the figure below, $$ABCD$$ and $$DEFG$$ are parallelograms. Point $$D$$ divides the segment $$AG$$ in the ratio $$1 : 2$$. Point $$E$$ is the midpoint of the segment $$DC$$. If $$BD$$ is perpendicular to $$DF$$, and $$\cos{\left( \angle{ADC} \right)} = 0.3$$, find the ratio $$AD : DE$$.