could someone help me with the followibg vecrot questions please:
1) Let A, B, C and D be the points with coordinates (1,3,4),(2,0,7),(3,5,−1) and (5,−1,5) respetively.
a) Write down the vector AB in the form ai + bj + ck.
b) Find the coordinates of the point E which is on the line AB between A and B and twice as far from A as from B.
c) Find the coordinates of another point on the line AB which is twice as far from A as from B.
d) Find the coordinates of the point H which divides the line BD in the ratio 2:3.
2) .For each of the following planes, give a vector equation of the form,
x = a(s,t)i y= b(s,t)j z=c(s,t)k,
where a(s,t),b(s,t),c(s,t) are expressions involving the parameters s and t, and hence parametric equations of the form,
x = a(s,t),y = b(s,t) and z = c(s,t), where x = xi+yj+zk.
a)The plane through the points with coordinates (3, −1, 5) and (2, 8, −6) and parallel to the vector
b)The plane containing the triangle ABC, where A, B and C have position vectors a = 3i − 4j + 2k,
b = 8i+9j−4k and c = 4i−3k;
3) Does the line through the point with coordinates (3, −5, 1) in the direction of the vector i + j − k meet the plane through the points with coordinates (3,−1,5) and (2,8,−6) and parallel to the vector 8i − j + 6k? If so, find the point(s) of intersection.
4 A rhombus is a parallelogram, all of whose sides have the same length,Show that the diagonals are perpendicular.
thank you !!!