Advertisements
Advertisements
प्रश्न
Show that the vector \[\hat{i} + \hat{j} + \hat{k}\] is equally inclined to the coordinate axes.
Advertisements
उत्तर
\[\text{ Let } \theta_1 \text{ be the angle between } \vec{a} \text{ and } x - axis.\]
\[\left| \vec{a} \right| = \sqrt{\left( 1 \right)^2 + \left( 1 \right)^2 + \left( 1 \right)^2} = \sqrt{3}\]
\[ \vec{b} = \hat{i} .........................\text{ (Because } \ \hat{i} \text{ is the unit vector along }x-axis)\]
\[\left| \vec{b} \right| = \sqrt{\left( 1 \right)^2} = \sqrt{1} = 1\]
\[ \vec{a} . \vec{b} = 1 + 0 + 0 = 1\]
\[\cos \theta_1 = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} = \frac{1}{\left( \sqrt{3} \right)\left( 1 \right)} = \frac{1}{\sqrt{3}}\]
\[ \Rightarrow \theta_1 = \cos^{- 1} \left( \frac{1}{\sqrt{3}} \right) . . . \left( 1 \right)\]
\[\]
\[\text{ Let } \theta_2 \text{ be the angle between } \vec{a} \ \text{ and } y - \text{ axis }\]
\[\left| \vec{a} \right| = \sqrt{\left( 1 \right)^2 + \left( 1 \right)^2 + \left( 1 \right)^2} = \sqrt{3}\]
\[ \vec{b} = \hat{j} ........................... \text { (Because } \hat{j}\ \text{ is the unit vector along }y-\text{ axis) }\]
\[\left| \vec{b} \right| = \sqrt{\left( 1 \right)^2} = \sqrt{1} = 1\]
\[ \vec{a} . \vec{b} = 0 + 1 + 0 = 1\]
\[\cos \theta_2 = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} = \frac{1}{\left( \sqrt{3} \right)\left( 1 \right)} = \frac{1}{\sqrt{3}}\]
\[ \Rightarrow \theta_2 = \cos^{- 1} \left( \frac{1}{\sqrt{3}} \right) . . . \left( 2 \right)\]
\[\text{ Let } \theta_3 \text{ be the angle between } \vec{a} \text{ and } z - \text{ axis }.\]
\[\left| \vec{a} \right| = \sqrt{\left( 1 \right)^2 + \left( 1 \right)^2 + \left( 1 \right)^2} = \sqrt{3}\]
\[ \vec{b} = \hat{k}.................................. { (Because } \ \hat{k}\ \text{ is the unit vector along }z-\text{ axis })\]
\[\left| \vec{b} \right| = \sqrt{\left( 1 \right)^2} = \sqrt{1} = 1\]
\[ \vec{a} . \vec{b} = 0 + 0 + 1 = 1\]
\[\cos \theta = \frac{\vec{a} . \vec{b}}{\left| \vec{a} \right| \left| \vec{b} \right|} = \frac{1}{\left( \sqrt{3} \right)\left( 1 \right)} = \frac{1}{\sqrt{3}}\]
\[ \Rightarrow \theta = \cos^{- 1} \left( \frac{1}{\sqrt{3}} \right) . . . \left( 3 \right)\]
\[\text{ From } (1), (2) \text{ and } (3), \text{ the given vector is equally inclined to the coordinate axes }.\]
APPEARS IN
संबंधित प्रश्न
Find the direction ratios of a vector perpendicular to the two lines whose direction ratios are -2, 1, -1, and -3, -4, 1.
If `bara, barb, bar c` are the position vectors of the points A, B, C respectively and ` 2bara + 3barb - 5barc = 0` , then find the ratio in which the point C divides line segment AB.
Find the value of 'p' for which the vectors `3hati+2hatj+9hatk and hati-2phatj+3hatk` are parallel
Represent graphically a displacement of 40 km, 30° east of north.
Two vectors having the same magnitude are collinear.
Two collinear vectors having the same magnitude are equal.
Show that the vector `hati + hatj + hatk` is equally inclined to the axes OX, OY, and OZ.
Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of the x-axis.
Find the value of x for which `x(hati + hatj + hatk)` is a unit vector.
If θ is the angle between two vectors `veca` and `vecb`, then `veca . vecb >= 0` only when ______.
Let `veca` and `vecb` be two unit vectors, and θ is the angle between them. Then `veca + vecb` is a unit vector if ______.
Find a vector of magnitude 4 units which is parallel to the vector \[\sqrt{3} \hat{i} + \hat{j}\]
Express \[\vec{AB}\] in terms of unit vectors \[\hat{i}\] and \[\hat{j}\], when the points are A (4, −1), B (1, 3)
Find \[\left| \vec{A} B \right|\] in each case.
ABCD is a parallelogram. If the coordinates of A, B, C are (−2, −1), (3, 0) and (1, −2) respectively, find the coordinates of D.
Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\] where \[\vec{a} = \hat{i} - \hat{j} \text{ and } \vec{b} = \hat{j} + \hat{k}\]
Find the angle between the vectors \[\vec{a} \text{ and } \vec{b}\] \[\vec{a} = 3\hat{i} - 2\hat{j} - 6\hat{k} \text{ and } \vec{b} = 4 \hat{i} - \hat{j} + 8 \hat{k}\]
Find a unit vector parallel to the vector \[\hat{i} + \sqrt{3} \hat{j}\]
Find the angles which the vector \[\vec{a} = \hat{i} -\hat {j} + \sqrt{2} \hat{k}\] makes with the coordinate axes.
If \[\left| \vec{a} + \vec{b} \right| = 60, \left| \vec{a} - \vec{b} \right| = 40 \text{ and } \left| \vec{b} \right| = 46, \text{ find } \left| \vec{a} \right|\]
For any two vectors \[\vec{a} \text{ and } \vec{b}\] show that \[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 0 \Leftrightarrow \left| \vec{a} \right| = \left| \vec{b} \right|\]
If either \[\vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0}\] then \[\vec{a} \cdot \vec{b} = 0 .\] But the converse need not be true. Justify your answer with an example.
If the vertices A, B and C of ∆ABC have position vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), respectively, what is the magnitude of ∠ABC?
If \[\overrightarrow{AO} + \overrightarrow{OB} = \overrightarrow{BO} + \overrightarrow{OC} ,\] prove that A, B, C are collinear points.
if `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2 hat"j" - 3hat"k" and hat"i" - 6hat"j" - hat"k"` respectively are the position vectors A, B, C and D, then find the angle between the straight lines AB and CD. Find whether `vec"AB" and vec"CD"` are collinear or not.
If `vec"a"` and `vec"b"` are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 1.5 BA.
The unit normal to the plane 2x + y + 2z = 6 can be expressed in the vector form as
The altitude through vertex C of a triangle ABC, with position vectors of vertices `veca, vecb, vecc` respectively is:
Area of rectangle having vertices A, B, C and D will position vector `(- hati + 1/2hatj + 4hatk), (hati + 1/2hatj + 4hatk) (hati - 1/2hatj + 4hatk)` and `(-hati - 1/2hatj + 4hatk)` is
Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2
If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.
In the vector \[\vec{AB}\], what are A and B respectively?
In three-dimensional geometry, what is the vector drawn from \[O(0,0,0)\] to \[P(x,y,z)\] called?
Which expression gives the magnitude of the position vector \[\vec{OP}\] for \[P(x,y,z)\]?
Which notation is the position vector of point \[P(x,y,z)\]?
