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Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.
If 
is a unit vector in the XY-plane, then 
Here, θ is the angle made by the unit vector with the positive direction of the x-axis.
Therefore, for θ = 30°:
Hence, the required unit vector is
.
Find the scalar components and magnitude of the vector joining the points
.
The vector joining the pointscan be obtained by,
Hence, the scalar components and the magnitude of the vector joining the given points are respectively 
and
A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.
Let O and B be the initial and final positions of the girl respectively.
Then, the girl’s position can be shown as:
Now, we have:
By the triangle law of vector addition, we have:
Hence, the girl’s displacement from her initial point of departure is
.
If
, then is it true that
? Justify your answer.
Now, by the triangle law of vector addition, we have.
It is clearly known that 
represent the sides of ΔABC.
Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side.
Hence, it is not true that.
Find the value of x for which
is a unit vector.
is a unit vector if
.
Hence, the required value of x is
.
Find a vector of magnitude 5 units, and parallel to the resultant of the vectors
.
We have,
Let
be the resultant of
.
Hence, the vector of magnitude 5 units and parallel to the resultant of vectors is
If
, find a unit vector parallel to the vector
.
We have,
Hence, the unit vector alongis
Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.
The given points are A (1, –2, –8), B (5, 0, –2), and C (11, 3, 7).
Thus, the given points A, B, and C are collinear.
Now, let point B divide AC in the ratio
. Then, we have:
On equating the corresponding components, we get:
Hence, point B divides AC in the ratio
Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are
externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.
It is given that
.
It is given that point R divides a line segment joining two points P and Q externally in the ratio 1: 2. Then, on using the section formula, we get:
Therefore, the position vector of point R is
.
Position vector of the mid-point of RQ =
Hence, P is the mid-point of the line segment RQ.
The two adjacent sides of a parallelogram are
and 
.
Find the unit vector parallel to its diagonal. Also, find its area.
Adjacent sides of a parallelogram are given as: 
and
Then, the diagonal of a parallelogram is given by
.
Thus, the unit vector parallel to the diagonal is
Area of parallelogram ABCD =
Hence, the area of the parallelogram is
square units.
Show that the direction cosines of a vector equally inclined to the axes OX, OY and OZ are
.
Let a vector be equally inclined to axes OX, OY, and OZ at angle α.
Then, the direction cosines of the vector are cos α, cos α, and cos α.
Hence, the direction cosines of the vector which are equally inclined to the axes are.
Let 
and
. Find a vector 
which is perpendicular to both 
and
, and
.
Let
.
Since
is perpendicular to both
and
, we have:
Also, it is given that:
On solving (i), (ii), and (iii), we get:
Hence, the required vector is
.
The scalar product of the vector
with a unit vector along the sum of vectors 
and 
is equal to one. Find the value of
.
Therefore, unit vector along
is given as:
Scalar product of
with this unit vector is 1.
Hence, the value of λ is 1.
If 
are mutually perpendicular vectors of equal magnitudes, show that the vector 
is equally inclined to 
and
.
Since
are mutually perpendicular vectors, we have
It is given that:
Let vector 
be inclined to at angles 
respectively.
Then, we have:
Now, as, 
.
Hence, the vector
is equally inclined to.
If θ is the angle between two vectors 
and 
, then 
only when
(A) 
(B) 
(C) 
(D) 
Let θ be the angle between two vectors and.
Then, without loss of generality, and are non-zero vectors so that
.
It is known that
.
Hence, when
.
The correct answer is B.
Let 
and 
be two unit vectors andθ is the angle between them. Then 
is a unit vector if
(A) 
(B) 
(C) 
(D) 
Let and be two unit vectors andθ be the angle between them.
Then, 
.
Now, is a unit vector if
.
Hence, is a unit vector if
.
The correct answer is D.
If θ is the angle between any two vectors 
and
, then 
when θisequal to
(A) 0 (B) 
(C) 
(D) π
Let θ be the angle between two vectors and.
Then, without loss of generality, and are non-zero vectors, so that
.
Hence, when θisequal to
.
The correct answer is B.
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, if and only if 
are perpendicular, given
. 
is
