Advertisements
Advertisements
प्रश्न
Express the vector `bar"a" = 5hat"i" - 2hat"j" + 5hat"k"` as a sum of two vectors such that one is parallel to the vector `bar"b" = 3hat"i" + hat"k"` and other is perpendicular to `bar"b"`.
उत्तर
Let `bar"a" = bar"c" + bar"d"`, where `bar"c"` is parallel to `bar"b" and bar"d"` is perpendicular to `bar"b"`.
Since, `bar"c"` is parallel to `bar"b", bar"c" = "m"bar"b"`, where m is a scalar.
∴ `bar"c" = "m"(3hat"i" + hat"k")`
i.e. `bar"c" = 3"m"hat"i" + "m"hat"k"`
Let `bar"d" = "x"hat"i" + "y"hat"j"+ zhat"k"`
Since, `bar"d"` is perpendicular to `bar"b" = 3hat"i" + hat"k", bar"d".bar"b" = 0`
∴ `("x"hat"i" + "y"hat"j" + "z"hat"k").(3hat"i" + hat"k") = 0`
∴ 3x + z = 0
∴ z = - 3x
∴ `bar"d" = "x"hat"i" + "y"hat"k" - 3"x"hat"k"`
Now, `bar"a" = bar"c" + bar"d"` gives
∴ `5hat"i" - 2hat"j" + 5hat"k" = (3"m"hat"i" + "m"hat"k") + ("x"hat"i" + "y"hat"j" - 3"x"hat"k")`
`= (3"m" + "x")hat"i" + "y"hat"j" + ("m" - 3"x")hat"k"`
By equality of vectors
3m + x = 5 ....(1)
y = - 2
and m - 3x = 5 ......(2)
From (1) and (2)
3m + x = m - 3x
∴ 2m = - 4x
∴ m = - 2x
Substituting m = - 2x in (1), we get
∴ - 6x + x = 5
∴ - 5x = 5
∴ x = - 1
∴ m = - 2x = 2
∴ `bar"c" = 6hat"i" + 2hat"k"` is parallel to `bar"b" and bar"d" = - hat"i" - 2hat"j" + 3hat"k"` is perpendicular to `bar"b"`
Hence, `bar"a" = bar"c" + bar"d", "where" bar"c" = 6hat"i" + 2hat"k" and bar"d" = - hat"i" - 2hat"j" + 3hat"k"`
APPEARS IN
संबंधित प्रश्न
If \[\vec{a}\], \[\vec{b}\], \[\vec{c}\] are position vectors of the vertices A, B and C respectively, of a triangle ABC, write the value of \[\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{CA} .\]
Write the position vector of a point dividing the line segment joining points A and B with position vectors \[\vec{a}\] and \[\vec{b}\] externally in the ratio 1 : 4, where \[\overrightarrow{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k} \text{ and }\overrightarrow{b} = - \hat{i} + \hat{j} + \hat{k} .\]
If \[\overrightarrow{a} = \hat{i} + \hat{j} , \vec{b} = \hat{j} + \hat{k} \text{ and }\vec{c} = \hat{k} + \hat{i} ,\] write unit vectors parallel to \[\overrightarrow{a} + \overrightarrow{b} - 2 \overrightarrow{c} .\]
If \[\overrightarrow{a} = \hat{i} + 2 \hat{j} - 3 \hat{k} \text{ and }\overrightarrow{b} = 2 \hat{i} + 4 \hat{j} + 9 \hat{k} ,\] find a unit vector parallel to \[\overrightarrow{a} + \overrightarrow{b}\].
If \[\vec{a} , \vec{b} , \vec{c}\] are three non-zero vectors, no two of which are collinear and the vector \[\vec{a} + \vec{b}\] is collinear with \[\vec{c} , \vec{b} + \vec{c}\] is collinear with \[\vec{a} ,\] then \[\vec{a} + \vec{b} + \vec{c} =\]
If G is the intersection of diagonals of a parallelogram ABCD and O is any point, then \[O \vec{A} + O \vec{B} + O \vec{C} + O \vec{D} =\]
If O and O' are circumcentre and orthocentre of ∆ ABC, then \[\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC}\] equals
The position vectors of the points A, B, C are \[2 \hat{i} + \hat{j} - \hat{k} , 3 \hat{i} - 2 \hat{j} + \hat{k}\text{ and }\hat{i} + 4 \hat{j} - 3 \hat{k}\] respectively.
These points
If OACB is a parallelogram with \[\overrightarrow{OC} = \vec{a}\text{ and }\overrightarrow{AB} = \vec{b} ,\] then \[\overrightarrow{OA} =\]
Show that the four points having position vectors
\[6 \hat { i} - 7 \hat { j} , 16 \hat {i} - 19 \hat {j}- 4 \hat {k} , 3 \hat {j} - 6 \hat {k} , 2 \hat {i} + 5 \hat {j} + 10 \hat {k}\] are not coplanar.
In the triangle PQR, `bar"PQ" = bar"2a", bar"QR" = bar"2b"`. The midpoint of PR is M. Find the following vectors in terms of `bar"a"` and `bar"b"`:
(i) `bar"PR"` (ii) `bar"PM"` (iii) `bar"QM"`.
If `|bara|` = 3, `|barb|` = 5, `|barc|` = 7 and `bara + barb + barc = bar0`, then the angle between `bara` and `barb` is ______.
Select the correct option from the given alternatives:
Let α, β, γ be distinct real numbers. The points with position vectors `alphahat"i" + betahat"j" + gammahat"k", betahat"i" + gammahat"j" + alphahat"k", gammahat"i" + alphahat"j" + betahat"k"`
Find the lengths of the sides of the triangle and also determine the type of a triangle:
L (3, -2, -3), M (7, 0, 1), N(1, 2, 1).
Find the unit vectors that are parallel to the tangent line to the parabola y = x2 at the point (2, 4).
If `bar"a", bar"b", bar"c"` are unit vectors such that `bar"a" + bar"b" + bar"c" = bar0,` then find the value of `bar"a".bar"b" + bar"b".bar"c" + bar"c".bar"a".`
Show that the vector area of a triangle ABC, the position vectors of whose vertices are `bar"a", bar"b" and bar"c"` is `1/2[bar"a" xx bar"b" + bar"b" xx bar"c" + bar"c" xx bar"a"]`.
Find the volume of the parallelopiped spanned by the diagonals of the three faces of a cube of side a that meet at one vertex of the cube.
lf `overlinea` and `overlineb` be two unit vectors and θ is the angle between them, then `|overlinea - overlineb|` is equal to ______
a and b are non-collinear vectors. If c = (x - 2)a + b and d = (2x + 1)a - b are collinear vectors, then the value of x = ______.
Using vectors, prove that cos (A – B) = cosA cosB + sinA sinB.
The 2 vectors `hat"j" + hat"k"` and `3hat"i" - hat"j" + 4hat"k"` represents the two sides AB and AC, respectively of a ∆ABC. The length of the median through A is ______.
Find the unit vector in the direction of the sum of the vectors `vec"a" = 2hat"i" - hat"j" + hat"k"` and `vec"b" = 2hat"j" + hat"k"`.
Using vectors, find the value of k such that the points (k, – 10, 3), (1, –1, 3) and (3, 5, 3) are collinear.
If `vec"a", vec"b", vec"c"` are unit vectors such that `vec"a" + vec"b" + vec"c"` = 0, then the value of `vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a"` is ______.
If `vec"a"` is any non-zero vector, then `(vec"a" .hat"i")hat"i" + (vec"a".hat"j")hat"j" + (vec"a".hat"k")hat"k"` equals ______.
Classify the following measures as scalar and vector.
2 meters north-west
Classify the following measures as scalar and vector.
10-19 coulomb
Classify the following measures as scalar and vector.
20 m/s2
Classify the following as scalar and vector quantity.
Velocity
`bara, barb` and `barc` are three vectors such that `veca + vecb + vecc` 20, `|bara| = 1, |barb| = 2` and `|barc| = 3`. Then `bara. barb + barb.barc + bar(c.a)` is equal to
The unit vector perpendicular to the vectors `6hati + 2hatj + 3hatk` and `3hati - 6hatj - 2hatk` is
Find `|veca xx vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`
Unit vector along `vec(PQ)`, where coordinates of P and Q respectively are (2, 1, – 1) and (4, 4, – 7), is ______.
Check whether the vectors `2hati + 2hatj +3hatk, - 3hati + 3hatj + 2hatk and 3hati + 4hatk` form a triangle or not.
In the triangle PQR, `bar(PQ) = 2bara and bar(QR) = 2barb`. The mid-point of PR is M. Find the following vectors in terms of `bara and barb`.
- `bar(PR)`
- `bar(PM)`
- `bar(QM)`
Consider the following statements and choose the correct option:
Statement 1: If `veca` and `vecb` represents two adjacent sides of a parallelogram then the diagonals are represented by `veca + vecb` and `veca - vecb`.
Statement 2: If `veca` and `vecb` represents two diagonals of a parallelogram then the adjacent sides are represented by `2(veca + vecb)` and `2(veca - vecb)`.
Which of the following is correct?
Check whether the vectors `2hati + 2hatj + 3hatk, -3hati + 3hatj + 2hatk and 3hati + 4hatk` form a triangle or not.
