ite an example of each of the following (assuming it is in 3-dimensional space).(5 marks) A point lying on thex-axis. A point lying on theyzplane. A point lying on both thexyandxzplanes. A point lying...

ite an example of each of the following (assuming it is in 3-dimensional space).(5 marks)

1. A point lying on thex-axis.
   
   


2. A point lying on theyzplane.
   
   


3. A point lying on both thexyandxzplanes.
   
   


4. A point lying on all three planes.
   
   


5. A point lying on none of the three planes, but equidistant from thexzandyzplanes.
   
   

TriangleABChas verticesA(4, 7, 7),B(1, 6, 5), andC(–2, 9, 8). What kind of triangle is ΔABC? Justify your answer.(4 marks)

The points (–2, 4, 5), (4, 5, –3), and (1, –1, 6) are three of four vertices of parallelogramABCD. Explain why there are three possibilities for the location of the fourth vertex, and find the three points.(4 marks)

The pointsA(–2, –1,z),B(2, 4, 3), andC(10,y, –1) are collinear. Find the values ofyandz.(2 marks)

Explainthe meaning of direction angles and their relation to direction vectors.(6 marks)

1. What are the direction angles of the vector [–5, 1, 8]?
   
   


2. If a pointPlies on thez-axis, what are the direction angles of the position vectorOP⇀" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-indent: 0px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">?
   
   


3. Prove thatcos2(α)+cos2(β)+cos2(γ)=1" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-indent: 0px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">.


4. Avector has direction anglesα = 85°" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-indent: 0px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">andβ= 65°" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-indent: 0px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">
   
   
   1. Find the value ofγ" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-indent: 0px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">.
   
   
   2. Find a vector that has those direction angles.
   
   
   
   


5. Explain why it is not possible for two of a vector's direction angles to be less than 45°.
   
   


6. What is the value ofsin2(α)+sin2(β)+sin2(γ)" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-indent: 0px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">? Why?
   
   

Explain the meanings of the terms linearly dependent and coplanar. Make sure you demonstrate that you understand the difference between the terms, and the situation in which linear dependency implies coplanarity.(4 marks)

Determine if the vectors [2, 4, –1], [8, –10, 5], and [5, –3, 2] are coplanar.(4 marks)

Giveexamples of sets of three vectors that are(3 marks)

1. Collinear


2. Coplanar


3. Not coplanar
   Explain your reasoning.

Explainhow you would prove if four given points are coplanar. Use your method to determine ifA(3, 4, –2),B(8, 5, 0),C(1, 10, –6), andD(9, 2, 2) are coplanar.(4 marks)

Determine if the following vectors are coplanar. Assume thatv1⇀" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-indent: 0px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">,v2⇀" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-indent: 0px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">, andv3⇀" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-indent: 0px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">are not coplanar.(4 marks)

w1⇀=2v1⇀+7v2⇀w2⇀=v2⇀+2v3⇀w3⇀=−v1⇀−7v3⇀" role="presentation" style="position: relative; display: inline-block; line-height: normal; font-size: 16px; text-align: left; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px;">