Vector Algebra

Step 1: Understand the relation between direction ratios and direction cosines.
The direction ratios of a line are proportional to the direction cosines. Let the direction ratios be , , and . The direction cosines , , and are given by the following relations:

Step 2: Calculate the magnitude of the direction ratios.
First, calculate the magnitude of the direction ratios:

Step 3: Calculate the direction cosines.
Now, calculate the direction cosines:

Step 4: Conclusion.
Thus, the direction cosines of the line are .

Step 1: General equation of a line.
The parametric form of the equation of a line passing through a point and parallel to the vector is given by:

Step 2: Apply the given data.
The line passes through the point and is parallel to the vector . Thus, the direction ratios are , , and . The equation of the line becomes:

Step 3: Conclusion.
The parametric equations for the line are: where is the parameter.

Step 1: Understand the concept of a unit vector.
A unit vector has a magnitude of 1. Therefore, we need to find the value of such that the magnitude of the vector is equal to 1.

Step 2: Find the magnitude of the vector.
The magnitude of the vector is: Now, the magnitude of is:

Step 3: Set the magnitude equal to 1.
For the vector to be a unit vector, we set the magnitude equal to 1: Solving for : Thus, the value of is:

Step 4: Conclusion.
Therefore, the value of for which the vector is a unit vector is .

Step 1: Formula for projection.
The formula to find the projection of vector on vector is given by:

Step 2: Compute dot product .
The dot product of and is:

Step 3: Compute magnitude of .
The magnitude of is:

Step 4: Find projection.
Now, substitute these values into the projection formula: Thus, the projection of on is:

Step 5: Conclusion.
Therefore, the projection of on is .

Step 1: Understanding the dot product.
The dot product of two vectors and is given by the formula: where is the angle between the two vectors.

Step 2: Apply the Cauchy-Schwarz inequality.
By the Cauchy-Schwarz inequality, we know: This is because lies in the range , so the magnitude of the dot product can never exceed the product of the magnitudes of the vectors.

Step 3: Conclusion.
Thus, we have shown that for any two vectors and , it always holds that:

Step 1: Definition of direction cosines.
The direction cosines of the coordinate axes , , and are the cosines of the angles that the respective axes make with the coordinate axes. The direction cosines of the -axis, -axis, and -axis are:

Step 2: Conclusion.
Thus, the direction cosines of the , , and -axes are 1, 0, and 0 respectively.