Given two equations of the line
in the vector form.
So, we will apply the distance formula for knowing the distance between two lines and
After comparing the given equations, we obtain
Then calculating the determinant value numerator.
That implies,
Now, after substituting the value in the above formula we get,
Therefore, units are the shortest...

Given two equations of line
in the vector form.
So, we will apply the distance formula for knowing the distance between two lines and
After comparing the given equations, we obtain
Then calculating the determinant value numerator.
That implies,
Now, after substituting the value in the above formula we get,
Therefore, is the...

We have given two lines:
and
Calculating the shortest distance between the two lines,
and
by the formula
Now, comparing the given equations, we obtain
Then calculating determinant
Now calculating the denominator,
So, we will substitute all the values in the formula above to obtain,
Since distance is always non-negative, the distance between the given lines is
units.

So given equation of lines;
and in the vector form.
Now, we can find the shortest distance between the lines and , is given by the formula,
Now comparing the values from the equation, we obtain
Then calculating
So, substituting the values now in the formula above we get;
Therefore, the shortest distance between the two lines is units.

First, we have to write the given equation of lines in the standard form;
and
Then we have the direction ratios of the above lines as;
and respectively..
Two lines with direction ratios and are perpendicular to each other if,
Therefore the two lines are perpendicular to each other.

First we have to write the given equation of lines in the standard form;
and
Then we have the direction ratios of the above lines as;
and respectively..
Two lines with direction ratios and are perpendicular to each other if,
Thus, the value of p is .

Given lines are;
and
So, we two vectors which are parallel to the pair of above lines respectively.
and
To find the angle A between the pair of lines we have the formula;
Then we have
and
Therefore we have;
or

To find the angle A between the pair of lines we have the formula;
We have two lines :
and
The given lines are parallel to the vectors ;
where and respectively,
Then we have
and
Therefore we have;
or

Let the line passing through the points and is AB;
Then as AB passes through through A so, we can write its position vector as;
Then direction ratios of PQ are given by,
Therefore the equation of the vector in the direction of AB is given by,
We have then the equation of line AB in vector form is given by,
So, the equation of AB in Cartesian form is;
or

GIven that the line is passing through the and
Thus the required line passes through the origin.
its position vector is given by,
So, the direction ratios of the line through and are,
The line is parallel to the vector given by the equation,
Therefore the equation of the line passing through the point with position vector and parallel to is given by;
Now, the equation of the line...

Given the Cartesian equation of the line;
Here the given line is passing through the point .
So, we can write the position vector of this point as;
And the direction ratios of the line are 3, 7, and 2.
This implies that the given line is in the direction of the vector, .
Now, we can easily find the required equation of line:
As we know that the line passing through the position vector and...

Given a line which passes through the point (– 2, 4, – 5) and is parallel to the line given by the ;
The direction ratios of the line, are 3,5 and 6.
So, the required line is parallel to the above line.
Therefore we can take direction ratios of the required line as 3k, 5k, and 6k, where k is a non-zero constant.
And we know that the equation of line passing through the point and with...

Given that the line is passing through the point with position vector and is in the direction of the line .
And we know the equation of the line which passes through the point with the position vector and parallel to the vector is given by the equation,
So, this is the required equation of the line in the vector form.
Eliminating , from the above equation we obtain the equation in the...

We have given points where the line is passing through it;
Consider the line joining the points (4, 7, 8) and (2, 3, 4) is AB and line joining the points (– 1, – 2, 1) and (1, 2, 5)..is CD.
So, we will find the direction ratios of the lines AB and CD;
Direction ratios of AB are
or
Direction ratios of CD are
or .
Now, lines AB and CD will be parallel to each other if
Therefore we...

It is given that the line is passing through A (1, 2, 3) and is parallel to the vector
We can easily find the equation of the line which passes through the point A and is parallel to the vector by the known relation;
, where is a constant.
So, we have now,
Thus the required equation of the line.

We have given points where the line is passing through it;
Consider the line joining the points (1, – 1, 2) and (3, 4, – 2) is AB and line joining the points (0, 3, 2) and (3, 5, 6).is CD.
So, we will find the direction ratios of the lines AB and CD;
Direction ratios of AB are
or
Direction ratios of CD are
or .
Now, lines AB and CD will be perpendicular to each other...

GIven direction cosines of the three lines;
And we know that two lines with direction cosines and are perpendicular to each other, if
Hence we will check each pair of lines:
Lines ;
the lines are perpendicular.
Lines ;
the lines are perpendicular.
Lines ;
the lines are perpendicular.
Thus, we have all lines are mutually perpendicular to each other.

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