Practice Questions

How many octants do the three coordinate planes divide the space into?

A point has coordinates $(m+1, m-3, 5)$ and it lies in the ZX-plane. Calculate the value of $m$.

Name the three coordinate planes in a three-dimensional rectangular coordinate system.

Justify why any point with coordinates $(0, y, z)$ must lie on the YZ-plane.

List the signs of the coordinates (x, y, z) for a point located in the third octant (III) and the seventh octant (VII).

Critique the following statement: 'The octant in which a point lies is determined solely by the signs of its $x$ and $y$ coordinates.'

Formulate the general condition, using inequalities, for a point $P(x, y, z)$ to be located in the VI octant.

State the formula for the distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$. Also, write the formula for the distance of a point $Q(x, y, z)$ from the origin.

Examine which octant the point $(-4, 1, -7)$ lies in.

A point lies on the z-axis. What are its x-coordinate and y-coordinate?

What are the coordinates of the origin in three-dimensional space?

Calculate the distance of the point $P(3, -4, 12)$ from the origin.

Explain the sign convention for distances measured along the axes from the origin.

Identify the plane determined by the y-axis and the z-axis taken together.

Describe the general form of the coordinates for any point lying in the YZ-plane. Explain why the x-coordinate is zero.

Describe the step-by-step process to locate a point $P(a, b, c)$ in space, where $a, b, c$ are all positive real numbers. Start from the origin O.

Identify the octants in which the following points lie: (a) $(2, -4, 6)$ (b) $(-1, -2, 3)$ (c) $(5, -3, -1)$

Calculate the coordinates of a point on the y-axis which is at a distance of $\sqrt{13}$ units from the point $P(2, -3, 0)$.

Find the coordinates of the point that is the reflection of $(-5, 2, -8)$ in the YZ-plane.

Evaluate whether three distinct points A, B, and C can form a triangle if the distance $AB + BC$ is exactly equal to the distance $AC$. Justify your reasoning.

Formulate an equation that represents the set of all points $P(x, y, z)$ such that the sum of its perpendicular distances from the XY-plane and the YZ-plane is equal to its perpendicular distance from the ZX-plane.

The points $A(1, 2, 3)$, $B(-1, -2, -1)$, and $C(2, 3, 2)$ are three consecutive vertices of a parallelogram. Evaluate the coordinates of the fourth vertex, D.

Explain what the coordinates $(x, y, z)$ of a point P represent in terms of its perpendicular distances from the three coordinate planes.

Summarize the key components of a rectangular coordinate system in three dimensions. Include a description of the coordinate axes, coordinate planes, and the origin.

Demonstrate that the points $A(1, 1, 1)$, $B(-2, 4, 1)$, and $C(-1, 5, 5)$ form a right-angled triangle.

The centroid of a triangle $PQR$ is at the origin $(0, 0, 0)$. If the coordinates of $P$ and $Q$ are $(4, -2, 5)$ and $(-3, 1, 2)$ respectively, calculate the coordinates of vertex $R$.

Calculate the coordinates of a point on the z-axis that is equidistant from the points $P(1, 5, 7)$ and $Q(5, 1, -4)$.

Analyze if the triangle formed by the points $A(3, 5, -4)$, $B(-1, 1, 2)$, and $C(-5, -5, -2)$ is an isosceles triangle.

Calculate the ratio in which the YZ-plane divides the line segment formed by joining the points $(-2, 4, 7)$ and $(3, -5, 8)$.

Find the coordinates of the point that divides the line segment joining the points $P(-2, 3, 5)$ and $Q(1, -4, 6)$ in the ratio $2:3$ (i) internally and (ii) externally.

Justify conclusively whether the points $A(1, -1, 3)$, $B(2, -4, 5)$, and $C(5, -13, 11)$ are collinear or form the vertices of a triangle.

Propose the coordinates of a point in the third octant that is equidistant from the XY-plane and the ZX-plane, and justify your proposal.

Derive the coordinates of the point on the y-axis which is equidistant from the points $P(3, 1, 2)$ and $Q(-2, 5, 4)$.

Formulate and prove a theorem stating the relationship between the length of the main diagonal of a rectangular parallelopiped and the lengths of its three distinct edges meeting at a vertex.

Critique the method of using only the distance formula to prove that four given points form a square. What additional verification is necessary and why?

Find the equation of the set of points $P(x, y, z)$ such that the sum of the squares of its distances from the points $A(1, 1, 1)$ and $B(-1, -1, -1)$ is equal to 18. Analyze the geometric shape this equation represents.

A point $P(x, y, z)$ is such that its distance from the point $A(2, 0, 0)$ is twice its distance from the point $B(-1, 0, 0)$. Solve for the equation of the set of all such points $P$.

Design a problem to demonstrate that four given points form a rhombus but not a square. Create the coordinates for the four vertices, and then provide a complete justification by calculating side lengths and diagonal lengths, explaining why the properties of a rhombus are met but those of a square are not.

Derive the formula for the coordinates of the circumcenter of a right-angled triangle whose vertices are $A(a, 0, 0)$, $B(0, b, 0)$, and $C(0, 0, c)$. Justify your reasoning.

Demonstrate that the points $A(1, 0, 1)$, $B(2, 2, 0)$, $C(3, 0, -1)$, and $D(2, -2, 0)$ are the vertices of a rhombus. Analyze if this rhombus is also a square.

Explain the concept of octants in three-dimensional geometry. List the names of all eight octants and present the signs of the coordinates (x, y, z) for each octant in a table format.

Three consecutive vertices of a parallelogram are $A(1, 2, -1)$, $B(3, 6, 11)$, and $C(5, 1, 0)$. Calculate the coordinates of the fourth vertex $D$.

Recall the coordinates of a point on the y-axis at a distance of 7 units from the origin in the negative direction.

Design a set of coordinates for three distinct points A, B, and C that form an isosceles right-angled triangle with the right angle at vertex B. Justify your design using the distance formula.

A point P moves such that its distance from point $A(2, 0, 0)$ is always twice its distance from point $B(-1, 0, 0)$. Derive the equation of the locus of P and identify the geometric shape it represents. Justify your conclusion.