Vector in space
1. What are the properties of a vector in space.
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2.Find the length of segment AB, which has midpoint S at the origin of the coordinate axes and point A [6; -2; 3]
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3.Determine the coordinates of vertex C [c1; c2; c3] in triangle ΔABC, if the vertices and centroid are at: A [3;3;3], B [-2;1;2], T [0;-1;0]. Also find the perimeter of this triangle.
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4.On the z-axis find the point that is equidistant from points A [ -2;1;4] and B [3;0;1]
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5. In the parallelepiped ABCDEFGH the coordinates of points A[2;-3;1], B[3;-4;2], D[4;2;-3], E[5;3;4] are known. Calculate the coordinates of the remaining points C, F, H, G
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6.Prove that triangle ΔABC is isosceles and right-angled if: A[2;-4;9], B[-1;-4;5], C[6;-4;6]
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7.Given the vectors:
Determine whether they are linearly dependent.
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8.Determine whether the given four points lie in one plane. A[1;2;-1], B[0;1;5], C[-1;2;1], D[2;1;3]
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9.Calculate the area of triangle ΔABC whose vertices are at the points:
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10.Calculate the volume of the quadrangular pyramid ABCDV, given its vertices: A[2;0;0], B[0;3;0], D[0;0;6], V[2;3;8]
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11. Determine the length of the median in triangle ABC, if A[1;0;2], B[-2;1;-3], C[0;-1;1].
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12. Determine the distance between the midpoints of segments AB and CD if:
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13.Let A[1;2;8] and B[4;y;4], and |AB| = 5. What is y?
Solution:
The required point is B[4;2;4].
14. Given points K, L, M, N. Calculate the coordinates of point N if:
Solution:
The required point is N [-4;3;-1].
15.Given points A[4;-3;2], B[5;0;-1]. Determine numbers m, n so that point C[2;m;n] lies on vector AB.
Solution:
The required point is C[2;-9;8].
16.Determine the mutual position (relationship) of the vectors
Solution:
The given vectors are perpendicular to each other.
17. Given three points in space A, B, C. Calculate the measure of the angle
Solution:
The angle between the vectors is α = 60°.
18.Calculate the perimeter and area of triangle ABC, if its vertices are:
A[0;1;2], B[1;2;0], C[2;0;1]
Solution:
19. In triangle ABC the vertices A, B and centroid T are given. Determine the coordinates of vertex C if
A[3;3;3], B[-2;1;2], T[0;-1;0], C[x,y,z]
Solution:
The third vertex of ΔABC is C[-1;-6;-5].
20.Write vector u as a linear combination of vectors v and w if:
Solution:
