91Ó°ÊÓ

For \(A(-1,3)\) and \(B(2,5),\) draw a coordinate plane and place the points on the graph. a. Draw vectors \(\overrightarrow{A B}\) and \(\overrightarrow{B A}\), and give vectors in component form equivalent to each of these vectors. b. Determine \(|\overrightarrow{O A}|\) and \(|\overrightarrow{O B}|\) c. Calculate \(|\overrightarrow{A B}|\) and state the value of \(|\overrightarrow{B A}|\)

If \(*\) is an operation on a set, \(S\), the element \(x\), such that \(a^{*} x=a\), is called the identity element for the operation *. a. For the addition of numbers, what is the identity element? b. For the multiplication of numbers, what is the identity element? c. For the addition of vectors, what is the identity element? d. For scalar multiplication, what is the identity element?

A student writes \(2(1,0)+4(-1,0)=(-2,0)\) and then concludes that (1,0) and (-1,0) span \(R^{2} .\) What is wrong with this conclusion?

State whether each statement is true or false. Justify your decision. a. If two vectors have the same magnitude, then they are equal. b. If two vectors are equal, then they have the same magnitude. c. If two vectors are parallel, then they are either equal or opposite vectors. d. If two vectors have the same magnitude, then they are either equal or opposite vectors.

a. Write the vector \(\overrightarrow{O A}=(-1,2,4)\) using the standard unit vectors. b. Determine \(|\overrightarrow{O A}|\)

An airplane is flying at an airspeed of \(300 \mathrm{km} / \mathrm{h}\). Using a scale of \(1 \mathrm{cm}\) equivalent to \(50 \mathrm{km} / \mathrm{h}\), draw a velocity vector to represent each of the following: a. a speed of \(150 \mathrm{km} / \mathrm{h}\) heading in the direction \(\mathrm{N} 45^{\circ} \mathrm{E}\) b. a speed of \(450 \mathrm{km} / \mathrm{h}\) heading in the direction \(\mathrm{E} 15^{\circ} \mathrm{S}\) c. a speed of \(100 \mathrm{km} / \mathrm{h}\) heading in an easterly direction d. a speed of \(300 \mathrm{km} / \mathrm{h}\) heading on a bearing of \(345^{\circ}\)

For each of the following, state whether the quantity is a scalar or a vector and give a brief explanation why: height, temperature, weight, mass, area, volume, distance, displacement, speed, force, and velocity.

Locate the points \(A(4,-4,-2), B(-4,4,2),\) and \(C(4,4,-2)\) using coordinate axes that you construct yourself. Draw the corresponding rectangular box (prism) for each, and label the coordinates of its vertices.

Draw two vectors, \(\vec{a}\) and \(\vec{b}\), that do not have the same magnitude and are noncollinear. Using the vectors you drew, construct the following: a. \(2 \vec{a}\) b. \(3 \vec{b}\) c. \(-3 \vec{b}\) d. \(2 \vec{a}+3 \vec{b}\) e. \(2 \vec{a}-3 \vec{b}\)

a. Name three vectors with their tails at the origin and their heads on the \(z\) -axis. b. Are the vectors you named in part a. collinear? Explain. c. How would you represent a general vector with its head on the \(z\) -axis and its tail at the origin?