What went wrong when the king rode under the arch?

It was too low and knocked off his crown

How does the chain of blame unfold in the poem?

Each accused person blames another, ending with the king's own changes

How many coordinates locate a point in space and why?

Three coordinates \((x,y,z)\), being the perpendicular distances of the point from the three mutually perpendicular coordinate planes.

Into how many octants do the coordinate planes divide space?

Eight octants, formed by the three coordinate planes \(XY\), \(YZ\) and \(ZX\).

Give the coordinates of a point on the \(x\)-axis and in the \(YZ\)-plane.

A point on the \(x\)-axis is \((x,0,0)\); a point in the \(YZ\)-plane is \((0,y,z)\).

State the distance formula in three dimensions.

\(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\) for \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\).

Find the distance between \(P(1,-3,4)\) and \(Q(-4,1,2)\).

\(\sqrt{(-5)^2+4^2+(-2)^2}=\sqrt{25+16+4}=\sqrt{45}=3\sqrt5\) units.

Name the octant in which \((-3,1,2)\) lies.

With \(x 0,\ z>0\), the point lies in the second octant.

How do you test whether three points are collinear in space?

Compute the three pairwise distances; the points are collinear iff the sum of two of them equals the third.

Coordinates of the origin and the form of a point on the \(z\)-axis?

The origin is \((0,0,0)\); any point on the \(z\)-axis is \((0,0,z)\).

If the centroid of triangle \(ABC\) is \((1,1,1)\), with \(A(3,-5,7),\ B(-1,7,-6)\), find \(C\).

Using the centroid average for each coordinate gives \(C=(1,1,2)\).

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Chapter 11: Introduction to Three Dimensional Geometry — Class 11 Mathematics

Mathematics · 14 chapters

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Chapter 11: Introduction to Three Dimensional Geometry

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A point in three-dimensional space is located by:

Summary

Locating a point in space requires three mutually perpendicular coordinate axes — the \(x\)-, \(y\)- and \(z\)-axes — meeting at the origin \(O\). These determine three coordinate planes (the \(XY\), \(YZ\) and \(ZX\) planes), which divide space into eight octants. A point \(P\) is given by an ordered triplet \((x,y,z)\), where \(x,y,z\) are its perpendicular distances from the \(YZ\), \(ZX\) and \(XY\) planes respectively; there is a one-to-one correspondence between points and triplets. The origin is \((0,0,0)\); a point on the \(x\)-axis is \((x,0,0)\), and a point in the \(YZ\) plane is \((0,y,z)\). The signs of the coordinates fix the octant. The distance between \(P(x_1,y_1,z_1)\) and \(Q(x_2,y_2,z_2)\) is \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\); in particular the distance of \(Q\) from the origin is \(\sqrt{x_2^2+y_2^2+z_2^2}\). This formula is used to test collinearity, identify triangles and parallelograms, and find equations of point sets.

Coordinate axes and coordinate planes in spaceOctantsCoordinates of a point in spaceSign conventions and octant identificationDistance between two pointsApplications: collinearity and figures

Key terms

Coordinate axes in space

Three mutually perpendicular lines — the \(x\), \(y\) and \(z\)-axes — meeting at the origin.

Coordinate planes

The \(XY\), \(YZ\) and \(ZX\) planes determined by pairs of axes.

Octants

The eight parts into which the three coordinate planes divide space.

Coordinates of a point

An ordered triplet \((x,y,z)\) of perpendicular distances from the \(YZ\), \(ZX\), \(XY\) planes.

Origin

The point \((0,0,0)\) where the three axes intersect.

Distance formula

\(PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\).

Important questions

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Coordinate axes in space

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Three mutually perpendicular lines — the \(x\), \(y\) and \(z\)-axes — meeting at the origin.

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Practice quiz · Introduction to Three Dimensional Geometry

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Introduction to Three Dimensional Geometry

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