1 Crore+ students have signed up on EduRev. Have you? 
and be three nonzero 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
We have
Given magnitude of projection of
∴ The required vector is either,
The vector
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 α^{2} = 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 noncollinear unit vectors. If u = a – (a . b) b and v = a × b, then  v  is
We have
Let be vector parallel to line of intersection of planes P_{1} and P_{2}. Plane P_{1} is parallel to the vectors and that P_{2} is parallel to then the angle between vector and a given vector
Normal to plane P_{1} 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 l_{2} at a distance of from the point of intersection of l and l_{1} 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 ℓ_{1} 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 L_{1}, L_{2} are coplanar, therefore
Let be three vectors each of magnitude and the angle between each pair of them is is a nonzero vector perpendicular to is a nonzero 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 R^{3}, consider the planes P_{1} : y = 0 and P_{2} : x + z = 1. Let P_{3} be the plane, different from P_{1} and P_{2}, which passes through the intersection of P_{1} and P_{2}. If the distance of the point (0, 1, 0) from P_{3} is 1 and the distance of a point (α, β, γ) from P_{3} is 2, then which of the following relations is (are) true?
P_{3} : x + λy + z – 1 = 0
In R^{3}, 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 P_{1} : x + 2y – z + 1 = 0 and P_{2} : 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 P_{1}. 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 midpoint, 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 R^{3} 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
132 docs70 tests

Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 
132 docs70 tests





