Enrichment, Extension, and Application

1 point

Exercises

1. Triangle $$OAB$$ has a right angle at $$O$$ and sides $$OA = 8, OB = 6$$. $$M$$ is the midpoint of the side $$AB$$. The segment $$OC = 12$$ is perpendicular to the plane of the triangle. Find the measure of $$\angle{CMB}$$, to the nearest tenth of a degree.
2. Find all unit vectors perpendicular to $$(1, 2, 3)$$ that make equal angles with the unit vectors $$\hat{i}$$ and $$\hat{j}$$.
3. Let $$\vec{u}, \vec{v}$$ be two unit vectors. Let $$\alpha$$ be the angle between $$\vec{u}$$ and $$\hat{i}$$, and $$\beta$$ the angle between $$\vec{v}$$ and $$\hat{i}$$, as shown in the diagram.
1. Express $$\vec{u}$$ in terms of $$\vec{i}$$, $$\vec{j}$$, and $$\alpha$$. Express $$\vec{v}$$ in terms of $$\vec{i}$$, $$\vec{j}$$, and $$\beta$$.
2. Use the dot product of the vectors $$\vec{u}$$ and $$\vec{v}$$ to find a formula for $$\cos\left( \alpha - \beta \right)$$.
4. In the diagram, $$ABCD$$ is a square of side length $$24$$. $$M$$ is the midpoint of $$BC$$ and $$N$$ is situated on $$DC$$, dividing $$DC$$ in the ratio $$1 : 2$$. $$AM$$ and $$BN$$ intersect at point $$P$$. Since $$A, P, M$$ are collinear, $$\overrightarrow{AP} = m\overrightarrow{AM}$$ for some constant $$m$$; similarly, since $$B, P, N$$ are collinear, $$\overrightarrow{BP} = n\overrightarrow{BN}$$ for some constant $$n$$.
1. Find the constants $$m$$ and $$n$$. Hint: Choose a coordinate system with $$A$$ at the origin and $$AB$$ as the $$x$$-axis. Then, write an equation connecting $$\overrightarrow{AP}$$ and $$\overrightarrow{BP}$$.
2. Find the area of $$\triangle{PBM}$$.
5. Consider the vectors $$\vec{a}, \vec{b}$$, and $$\vec{c}$$. If $$\vec{u} = \left( \vec{a} \cdot \vec{b} \right) \vec{c} - \left( \vec{a} \cdot \vec{c} \right) \cdot \vec{b}$$, prove that $$\vec{a}$$ is perpendicular to $$\vec{u}$$.
6. Triangle $$OAB$$ is inscribed in a circle with centre at $$C$$. Points $$O~(0, 0)$$, $$A~(2a, 2b)$$, and $$B~(2c, 0)$$ are given.
1. Express the coordinates of $$C$$ in terms of $$a$$, $$b$$, and $$c$$.
2. Find a point $$H$$ such that $$\overrightarrow{CH} = \overrightarrow{CO} + \overrightarrow{CA} + \overrightarrow{CB}$$.
3. Prove that all three altitudes of $$\triangle{ABC}$$ pass through $$H$$ by showing that $$\overrightarrow{OH} \perp \overrightarrow{AB}$$, $$\overrightarrow{BH} \perp \overrightarrow{OA}$$, and $$\overrightarrow{AH} \perp \overrightarrow{OB}$$.
7. Vectors $$\vec{u}, \vec{v}, \vec{w}$$ are perpendicular to each other and $$\left\lvert \vec{u} \right\rvert = 1, \left\lvert \vec{v} \right\rvert = 3, \left\lvert \vec{w} \right\rvert = 4$$. Find the magnitude of the vector $$\left( \vec{u} \times \vec{v} \right) + \left( \vec{v} \times \vec{w} \right) + \left( \vec{w} \times \vec{u} \right)$$.
8. Vectors $$\vec{u}$$ and $$\vec{v}$$ are perpendicular and $$\left\lvert \vec{u} \right\rvert = \sqrt{3} \left\lvert \vec{v} \right\rvert$$. Find all vectors $$\vec{x}$$ which satisfy $$\vec{u} \times \vec{x} = \vec{x} \times \vec{v}$$.
9. Let $$\vec{u} = (-1, 2, 0)$$, $$\vec{v} = (-1, 0, 1)$$, and $$\vec{w} = (0, 1, 1)$$.
1. Find $$\vec{z} = \left( \vec{u} \times \vec{v} \right) \times \left( \vec{u} \times \vec{w} \right)$$, and show that $$\vec{z}$$ is collinear with $$\vec{u}$$.
2. Explain why, for any three non-coplanar vectors $$\vec{u}$$, $$\vec{v}$$, and $$\vec{w}$$, the vector $$\vec{z} = \left( \vec{u} \times \vec{v} \right) \times \left( \vec{u} \times \vec{w} \right)$$ is collinear with $$\vec{u}$$.
10. Let $$\vec{u} = (1, 2, 3), \vec{v} = (-3, 1, 2)$$.
1. If $$\vec{w} = (-1, 5, 8)$$, calculate $$\left( \vec{u} \times \vec{v} \right) \cdot \vec{w}$$ and explain why $$\vec{u}$$, $$\vec{v}$$, and $$\vec{w}$$ are coplanar. Express $$\vec{w}$$ as a linear combination of the vectors $$\vec{u}$$ and $$\vec{v}$$.
2. If $$\vec{z} = (-1, 2, 0)$$, show that $$\vec{u}, \vec{v}, \vec{z}$$ are not coplanar and hence form a basis for the space. Express $$\vec{x} = (2, 3, 11)$$ as a linear combination of the vectors.
1. In the regular tetrahedron $$ABCD$$, $$M$$ is the midpoint of $$BC$$ and $$AH$$ is the altitude of triangle $$AMD$$. Prove that $$AH$$ is perpendicular to the plane containing $$\triangle{BCD}$$.
2. If the sides of tetrahedron $$ABCD$$ are length $$1$$, and $$B$$ is the origin of a coordinate system, with $$BC$$ lying on the $$x$$-axis, and $$\triangle{BCD}$$ in the first quadrant, write $$\overrightarrow{BA}$$, $$\overrightarrow{BC}$$, and $$\overrightarrow{BD}$$ as algebraic vectors.
3. Calculate the volume of the tetrahedron described in b).
1. Prove that if $$\overrightarrow{PA_1} + \overrightarrow{PA_2} + \cdots + \overrightarrow{PA_n} = \vec{0}$$, then $$\overrightarrow{OP} = \dfrac{\overrightarrow{OA_1} + \overrightarrow{OA_2} + \cdots + \overrightarrow{OA_n}}{n}$$.
2. Tetrahedral structures like the one in problem 9 can appear in nature as the methane molecule $$\text{CH}_4$$, where four hydrogen atoms surround a central carbon atom with tetrahedral symmetry.

In the following diagram, we have $$\overrightarrow{CH_1} + \overrightarrow{CH_2} + \overrightarrow{CH_3} + \overrightarrow{CH_4} = \vec{0}$$ and $$\left\lvert \overrightarrow{CH_1} \right\rvert = \left\lvert \overrightarrow{CH_2} \right\rvert = \left\lvert \overrightarrow{CH_3} \right\rvert = \left\lvert \overrightarrow{CH_4} \right\rvert$$. The tetrahedral bond angle is the angle formed between any two bonds; for instance, the angle $$H_1CH_2$$. Calculate this angle.

Note: $$\text{CH}_4$$ is a chemical formula describing the composition of the entire methane molecule; $$CH_4$$ is the line segment between the points $$C$$ and $$H_4$$ in the diagram, representing the bond between the carbon atom and that specific hydrogen atom.

1. Prove the theorem of Apollonius, which states:
In $$\triangle{ABC}$$, if $$M$$ is the midpoint of $$BC$$, then $$AB^2 + AC^2 = 2AM^2 + 2MC^2$$
Hint: Denote $$\overrightarrow{AB} = \vec{b}$$, $$\overrightarrow{AC} = \vec{c}$$, and use the dot product.
2. In $$\triangle{ABC}$$, $$AB = 8$$, $$AC = 11$$, and $$BC = 9$$. Find the length of the median $$AM$$.
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