#### If the direction ratios of the first line are 4, 3, 1 and the direction ratios of the second line are 1, 1, k and these two lines are perpendicular to each other, then the value of k is _________.

If the direction ratios of the first line are 4, 3, 1 and the direction ratios of the second line are 1, 1, k and these two lines are perpendicular to each other, then the value of k is -7. The direction ratios of the first line are a1, b1, c1 and the direction ratios of the second line are a2, b2, c2. If these two lines are perpendicular to each other, then a1a2 + b1b2 + c1c2 = 0 Here, a1 = 4, b1 = 3, c1 = 1 and a2 = 1, b2 = 1, c2 = k ⇒ 4 +3 + k = 0 ⇒ k = -7

#### __________plane/planes can be drawn if given the condition that it passes through two given non collinear points.

Many planes can be drawn given the condition that it passes through two given non collinear points. A plane is determined uniquely if it passes through three given non collinear points. If a plane passes through less than three non collinear points, then it cannot be unique i.e. many planes can be drawn through two non collinear points.

#### If two lines are parallel in space, then the shortest distance between them will be ________________.

If two lines are parallel in space, then the shortest distance between them will be the perpendicular distance, i.e. the length of the perpendicular drawn from a point on one line onto the other line.

#### To obtained a unique plane, a minimum _________ non-collinear point/s are required.

To obtain a unique plane, a minimum of three non-collinear points are required. Unique equation of plane can be obtained if it is passing through three or more than three non collinear points. Many equation of plane can be obtained if it is passing through one or two non collinear points.

#### Any three numbers which are proportional to the direction cosines of a line are called the ________ of the line.

By definition, any three numbers which are proportional to the direction cosines of a line are called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are direction ratios of a line, then a = kl, b = km and c = kn, for any nonzero k ∈ R.

#### The line of the shortest distance between two skew lines will be perpendicular to ____________line/lines.

The line of the shortest distance of two skew lines will be perpendicular to the both lines. By the shortest distance between two lines, we mean the join of a point on one line with one point on the other line so that the length of the segment so obtained is the smallest.

#### If three points are on the same line, then there will be __________plane / planes that will contain them.

A unique equation of plane can be obtained if it is passing through three non collinear points. And if these three point are collinear, then the planes passing through these point cannot be unique. Many planes can be constructed passing through these collinear points.

#### In space, a pair of lines which are neither intersecting nor parallel is called ___________lines.

In space, there are lines which are neither intersecting nor parallel. In fact, such pair of lines are non-coplanar and are called skew lines.

#### The direction cosines of two __________ lines have same set of direction cosines.

The direction cosines of two parallel lines have the same set of direction cosines. The direction cosines are l, m and n, where l = cos α, m = cos β and n = cos γ Hence, the direction cosines of two parallel directed lines have the same set of direction cosines.

#### If in space two lines intersect at a point, then the shortest distance between them is _____________unit/s.

if in space two lines intersect at a point, then the shortest distance between them is zero units. If two lines intersect at a point, then the intersecting point lies on both the lines, and hence the distance between the two lines is zero.