Question 1: Find the magnitude of the vector
Solution:
Magnitude of a vector
=>
=>
=>
=>
Question 2: Find the unit vector in the direction of
Solution:
We know that unit vector of a vector
is given by, =>
=>
=>
=>
=>
Question 3: Find a unit vector in the direction of the resultant of the vectors
Solution:
Let,
=>
=>
=>
Let
be the resultant, =>
=>
=>
Unit vector is,
=>
=>
=>
=>
Question 4: The adjacent sides of a parallelogram are represented by the vectors
Solution:
Let PQRS be the parallelogram.
Given that, PQ =
and QR = . Thus, the diagonals are: PR and SQ.
=>
=>
=>
=>
=>
=>
=>
=>
Thus the unit vectors in the direction of the diagonals are:
=>
=>
=>
=>
=>
=>
Question 5: If
Solution:
Given,
, and . Let,
=>
=>
=>
=>
The magnitude is given by,
=>
=>
=>
Question 6: If
Solution:
Given,
And,
=>
=>
=>
=>
=> Thus the coordinates of Q are (4,1,1).
Question 7: Prove that the points
Solution:
Let,
=>
=>
=>
Thus, the 3 sides of the triangle are,
=>
=>
=>
=>
=>
=>
=>
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The lengths of every side are given by their magnitude,
=>
=>
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As we can see,
=>
=> These 3 points form a right-angled triangle.
Question 8: If the vertices A, B and C of a triangle ABC are the points with position vectors
Solution:
Let,
=>
=>
=>
The sides of the triangle are given as,
=>
=>
=>
=>
=>
=>
=>
=>
=>
The lengths of the sides are,
=>
=>
=>
Question 9: Find the vector from the origin O to the centroid of the triangle whose vertices are (1,-1,2), (2,1,3), and (-1,2,-1).
Solution:
The position of the centroid is given by,
=> (x, y, z) =
=> (x, y, z) =
=> (x, y, z) =
The vector to the centroid from O is,
=>
Question 10: Find the position vector of a point R which divides the line segment joining points p(
(i) Internally
Solution:
The position vectors of a point that divides a line segment internally are given by,
=>
, where =>
=>
(ii) Externally
Solution:
The position vectors of a point that divides a line segment externally are given by,
=>
, where =>
=>