Points and vectors

1103040208

Level: 
C
We are given the points A=[4;5;−1], B=[−2;−1;2], C=[−1;−3;0] and D=[0;m;2]. Find the missing coordinate of the point D such that the point D lies in the plane determined by the points A, B and C. Hint: Use a linear combination of vectors shown in the picture or use their mixed product.
m=3
m=−3
m=1
m does not exist

1003040207

Level: 
C
Given the points A=[2;0;3] and B=[−1;2;0], specify all the points C lying on the z-axis, such that the area of the triangle ABC is 22. Hint: Use a cross product of vectors.
C1=[0;0;1]; C2=[0;0;2913]
C1=[0;0;1]; C2=[0;0;−1]
C1=[0;0;−1]; C2=[0;0;1329]
C1=[0;0;−1]; C2=[0;0;2913]

1103040206

Level: 
C
Given the points A=[1;5] and B=[−4;2], specify all the points C lying on the x-axis, such that the area of the triangle ABC is 14. Hint: Use a cross product of vectors.
C1=[2;0]; C2=[−503;0]
C1=[1;0]; C2=[−473;0]
C1=[2;0]; C2=[−473;0]
C1=[1;0]; C2=[−503;0]