Let u, v, and w be pairwise orthogonal vectors. (a) Show that

$$\| \mathbf { u } + \mathbf { v } + \mathbf { w } \| ^ { 2 } = \| \mathbf { u } \| ^ { 2 } + \| \mathbf { v } \| ^ { 2 } + \| \mathbf { w } \| ^ { 2 }$$

. (b) If u, v, and w are all the same length, show that they all make the same angle with u+v+w.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Even if I'm not sure how mathemat icians define irrational and complex numbers, telling me how these sets are related, I can construct a Venn diagram illustrating their relationship.

Explain how to use intercepts to graph the general form of a line's equation.

(This logic problem dates back to the eighth century.) A farmer needs to take his goat , wolf, and cabbage across a stream. His boat can hold him and one other passenger (the goat , wolf, or cabbage). If he takes the wolf with him , the goat will eat the cabbage. If he takes the cabbage, the wolf will eat the goat. Only when the farmer is present are the cabbage and goat safe from their respective predators. How does the farmer get everything across the stream?

Solve and check.

$$2 ^ { x } = 4 ^ { x + 1 }$$

Organize the divisions of the geologic time scale in order from longest length of time to shortest length of time.

For a polygon to be convex means that all of its interior angles are less than 180 degrees. Use mathematical induction to prove that for all integers

$$n \geq 3$$

, the angles of any n-sided convex polygon add up to 180(n-2) degrees.

Mike dove to an elevation of -40 feet, and Julio dove to an elevation of -50 feet. Mike says that because -40 is greater than -50, he dove farther than Julio. Use absolute value to explain Mike's error.

Simplify the complex fraction.

$$\frac { \frac { 3 } { b ^ { 2 } } - \frac { 4 } { b } + 1 } { 1 - \frac { 1 } { b } - \frac { 6 } { b ^ { 2 } } }$$

Solve each system by graphing. Tell whether the system has one solution, infinitely many solutions, or no solution.

$$\left. \begin{array} { l } { y = \frac { 4 } { 3 } x - 2 } \\ { 3 y - 4 x = - 6 } \end{array} \right.$$

Choose the best answer. Rhonda earns 5.5% commission when she sells a house. The amount of commission she receives from a sale is $18,700. What is the sale price of the house? A.$1028.50 B. $321,300 C.$340,000 D. $358,700

Solve each equation. 4 b-3=17

What is a prodigy?

The A locus and the D locus are so tightly linked that no recombination is ever observed between them. If Ad/ Ad is crossed with aD/aD and the

$$F_1$$

is intercrossed, what phenotypes will be seen in the

$$F_2$$

and in what proportions?