Advertisements
Advertisements
प्रश्न
Show that the vectors \[\vec{a} = \frac{1}{7}\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right), \vec{b} = \frac{1}{7}\left( 3\hat{i} - 6 {j} + 2 \hat{k} \right), \vec{c} = \frac{1}{7}\left( 6 \hat{i} + 2 \hat{j} - 3 {k} \right)\] are mutually perpendicular unit vectors.
Advertisements
उत्तर
\[\left| \vec{a} \right| = \frac{1}{7}\sqrt{2^2 + 3^2 + 6^2} = \frac{1}{7}\sqrt{49} = \frac{7}{7} = 1\]
\[\left| \vec{b} \right| = \frac{1}{7}\sqrt{3^2 + \left( - 6 \right)^2 + 2^2} = \frac{1}{7}\sqrt{49} = \frac{7}{7} = 1\]
\[\left| \vec{c} \right| = \frac{1}{7}\sqrt{6^2 + 2^2 + \left( - 3 \right)^2} = \frac{1}{7}\sqrt{49} = \frac{7}{7} = 1\]
\[\text{ And }\]
\[ \vec{a} . \vec{b} \]
\[ = \frac{1}{7}\left( 2 \hat{i} + 3 \hat{j} + 6 {k} \right) . \frac{1}{7}\left( 3 t{i} - 6 \hat{j} + 2 \hat{k} \right)\]
\[ = \frac{1}{49}\left( 6 - 18 + 12 \right)\]
\[ = 0\]
\[ \vec{b} . \vec{c} \]
\[ = \frac{1}{7}\left( 3 \hat{i} - 6 \hat{j} + 2 \hat{k} \right) . \frac{1}{7}\left( 6 \hat{i} + 2 \hat{j} - 3 \hat{k} \right)\]
\[ = \frac{1}{49}\left( 18 - 12 - 6 \right)\]
\[ = 0\]
\[ \vec{c} . \vec{a} \]
\[ = \frac{1}{7}\left( 6 \hat{i} + 2 \hat{j} - 3 \hat{k} \right) . \frac{1}{7}\left( 2 \hat{i} + 3 \hat{j} + 6 \hat{k} \right)\]
\[ = \frac{1}{49}\left( 12 + 6 - 18 \right)\]
\[ = 0\]
\[So,\left| \vec{a} \right| = \left| \vec{b} \right| = \left| \vec{c} \right| = 1\text{ and } \vec{a} . \vec{b} = \vec{b} . \vec{c} = \vec{c} . \vec{a} = 0\]
\[\text{ So }, \text{ the given vectors are mutually perpendicular unit vectors. }\]
APPEARS IN
संबंधित प्रश्न
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 position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector
`2hati+3hatj+4hatk` to the plane `vecr` . `(2hati+hatj+3hatk)−26=0` . Also find image of P in the plane.
Find the position vector of a point which divides the join of points with position vectors `veca-2vecb" and "2veca+vecb`externally in the ratio 2 : 1
If `bara, barb, barc` are position vectors of the points A, B, C respectively such that `3bara+ 5barb-8barc = 0`, find the ratio in which A divides BC.
Represent graphically a displacement of 40 km, 30° east of north.
Two collinear vectors are always equal in magnitude.
Two vectors having the same magnitude are collinear.
Two collinear vectors having the same magnitude are equal.
Find the direction cosines of the vector joining the points A (1, 2, -3) and B (-1, -2, 1) directed from A to B.
Show that the vector `hati + hatj + hatk` is equally inclined to the axes OX, OY, and OZ.
Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).
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 a unit vector parallel to the vector \[\hat{i} + \sqrt{3} \hat{j}\]
Find the angle between the vectors \[\vec{a} = \hat{i} + 2 \hat{j} - \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k}\]
If \[\vec{a,} \vec{b,} \vec{c}\] are three mutually perpendicular unit vectors, then prove that \[\left| \vec{a} + \vec{b} + \vec{c} \right| = \sqrt{3}\]
If \[\vec{\alpha} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \text{ and } \vec{\beta} = 2 \hat{i} + \hat{j} - 4 \hat{k} ,\] then express \[\vec{\beta}\] in the form of \[\vec{\beta} = \vec{\beta_1} + \vec{\beta_2} ,\] where \[\vec{\beta_1}\] is parallel to \[\vec{\alpha} \text{ and } \vec{\beta_2}\] is perpendicular to \[\vec{\alpha}\]
Show that the vectors \[\vec{a} = 3 \hat{i} - 2 \hat{j} + \hat{k} , \vec{b} = \hat{i} - 3 \hat{j} + 5 \hat{k} , \vec{c} = 2 \hat{i} + \hat{j} - 4 \hat{k}\] form a right-angled triangle.
If \[\vec{a} = 2 \hat{i} + 2 \hat{j} + 3 \hat{k} , \vec{b} = - \hat{i} + 2 \hat{j} + \hat{k} \text{ and } \vec{c} = 3 \hat{i} + \hat{j}\] \[\vec{a} + \lambda \vec{b}\] is perpendicular to \[\vec{c}\] then find the value of λ.
Find the magnitude of two vectors \[\vec{a} \text{ and } \vec{b}\] that are of the same magnitude, are inclined at 60° and whose scalar product is 1/2.
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?
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).
Show that the points \[A \left( 2 \hat{i} - \hat{j} + \hat{k} \right), B \left( \hat{i} - 3 \hat{j} - 5 \hat{k} \right), C \left( 3 \hat{i} - 4 \hat{j} - 4 \hat{k} \right)\] are the vertices of a right angled triangle.
Find the value of x for which \[x \left( \hat{i} + \hat{j} + \hat{k} \right)\] is a unit vector.
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.
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
Geometrically, a vector is represented by which of the following?
Which notation can be used for the magnitude of vector \[\vec{AB}\]?
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 \[P(x,y,z)\]?
What is the magnitude of the position vector of \[P(3,4,0)\]?
