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and be three non-zero vectors such that is a unit vector perpendicular to both the vectors and If the angle between then
The number of vectors of unit length perpendicular to vectors
We know that if then
as represent two vectors in opp. directions.
∴ We have two possible values of
Let be three vectors. A vector in the plane of whose projection on
Given magnitude of projection of
∴ The required vector is either,
If and are linearly dependent vectors and
Given that, and are linearly dependent,
NOTE THIS STEP :
for some scalars l and m not all zeros.
Þ l + 4m = 1 ...(1)
l + 3m = α ...(2)
l + 4m = β ...(3)
From (1) and (3) we have, β = 1
Also given that
Substituting the value of β we get α2 = 1
⇒ a = ±1
For three vectors u, v, w which of the following expression is not equal to any of the remaining three?
Which of the following expressions are meaningful?
As dot product of two vectors gives a scalar quantity.
Let a and b be two non-collinear unit vectors. If u = a – (a . b) b and v = a × b, then | v | is
Let be vector parallel to line of intersection of planes P1 and P2. Plane P1 is parallel to the vectors and that P2 is parallel to then the angle between vector and a given vector
Normal to plane P1 is
Now, angle between is given by
The vector (s) which is/are coplanar with vectors and and perpendicular to the vector is/are
∵ Required vector is coplanar with
∴a and d are the correct options
If the straight lines are coplanar, then the plane (s) containing these two lines is (are)
For given lines to be coplanar, we should have
For k = 2, obviously the plane y + 1 = z is common in both lines.
For k = – 2, the plane is given by
A line l passing through the origin is perpendicular to the lines
Then, the coordinate(s) of the point(s) on l2 at a distance of from the point of intersection of l and l1 is (are)
The given lines are
Let direction ratios of l be a, b, c then as
∴ a + 2b + 2c = 0
2a + 2b + c = 0
Any point on ℓ1 is (t + 3, 2t – 1, 2t + 4) and any point on ℓ is (2λ, –3λ, 2λ)
Two lines are coplanar. Then a can take value(s)
As L1, L2 are coplanar, therefore
Let be three vectors each of magnitude and the angle between each pair of them is is a non-zero vector perpendicular to is a non-zero vector perpendicular to
Angle between each pair is
(d) is not correct.
From a point P(λ, λ, λ), perpendicular PQ and PR are drawn respectively on the lines y = x, z = 1 and y = – x, z = – 1. If P is such that ∠QPR is a right angle, then the possible value(s) of ∠ is/(are)
Lines are x = y, z = 1
Let Q (a, a, 1) and R (– β,β, – 1) Direction ratios of PQ are λ – α, λ – α, λ – 1 and direction ratios of PR are λ + β, λ – β, λ + 1
∴ PQ is perpendicular to line (1)
∴ – (λ + β) + λ – β = 0 ⇒ b = 0
∴ dr’s of PQ are 0, 0, λ – 1 and dr’s of PR are λ, λ, λ + 1
∵ ∠QPR = 90° ⇒ (λ – l) (λ + 1) = 0 ⇒ λ = 1 or – 1
But for λ = 1, we get point Q itself
∴ we take λ = – 1
In R3, consider the planes P1 : y = 0 and P2 : x + z = 1. Let P3 be the plane, different from P1 and P2, which passes through the intersection of P1 and P2. If the distance of the point (0, 1, 0) from P3 is 1 and the distance of a point (α, β, γ) from P3 is 2, then which of the following relations is (are) true?
P3 : x + λy + z – 1 = 0
In R3, let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes P1 : x + 2y – z + 1 = 0 and P2 : 2x – y + z – 1 = 0. Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane P1. Which of the following points lie (s) on M?
For locus of M,
On checking the given point, we find and satisfy the above eqn.
Let ΔPQR be a triangle. then which of the following is (are) true?
Consider a pyramid OPQRS located in the first octant (x > 0, y > 0, z > 0) with O as origin, and OP and OR along the x–axis and the y–axis, respectively. The base OPQR of the pyramid is a square with OP = 3. The point S is directly above the mid-point, T of diagonal OQ such that TS = 3. Then
The coordinates of vertices of pyramid OPQRS will be
O(0, 0, 0), P (3, 0, 0), Q (3, 3, 0), R (0, 3, 0),
dr's of OQ = 1, 1, 0
dr's of OS = 1, 1, 2
∴ acute angle between OQ and OS
⇒ 2x – 2y = 0 or x – y = 0
length of perpendicular from P (3, 0, 0) to plane x – y = 0
If ON is perpendicular to RS, then N
Let be a unit vector in R3 and Given that there exists a vector in Which of the following statement(s) is (are) correct?
where α is the angle between
From (i) and (ii) cos α = 1 ⇒ a = 0°
is perpendicular to the plane containing
is perpendicular to
Clearly there can be infinite many choices for