c) What is the cosine of the angle between the planes x + 2y – z + 1 = 0, and -x+ 3z + 1= 0? d) Find a unit vector in R' in the opposite direction to v = [1, 2, -2].

Q: 2. Let f(x,y) = t Find the direction of a unit vector so that Daf is minimal at point (0, 1). %3D 3…

A: To find the direction of a unit vector u so that the directional derivative of f at point (0,1) in…

Q: For vectors i : onto v. and w = calculate the scalar projection of w %3D comp;w (Give an answer…

A: scalar projection of w on v is =v.w|v||w|compvw¯=…

Q: Evaluate | F. dS , where F = and S is the helicoid with vector equation r(u, v) = 0 < u < 4, 0 < v…

A:

Q: At how many point(s) does the helix with vector equation r(t) : (cos t, sin t, t) intersect the…

A:

Q: Give the unit vector in spherical coordinates that extends from M(5,1,2) to N(2,4,5).

A:

Q: If the projections of vector a on x,y and z −axes are 2,1,2 units respectively, find the angle at…

A:

Q: A force F = 2i -j- 3k is applied to an airplane with displacement vector D = i+2j + 2k. Express F as…

A: Let the force vector be F→=2i-j-3k and the displacement vector D→=i+2j+2k    F→=β1+β2   Here, β1 is…

Q: The angle between two unit vectors e and f is π/6 . What is the length of e × f?

A: The angle between two unit vectors e and f is π6.

Q: Prove that u = (cos θ)i − (sin θ)j and v = (sin θ)i + (cos θ)j                               are…

A: We have given;  u = (cos θ)i − (sin θ)j and v = (sin θ)i + (cos θ)j

Q: (a) Find the unit vector of 2q where q=(8 sin 0,-8 cos 0,0).

A:

Q: Express in cylindrical components: (a) the vector from C(3, 2, -7) to D(-1, -4, 2); (b) a unit…

A:

Q: Evaluate F.dS, where F = and S is the helicoid with vector equation r(u, v) = 0 < u < 3, 0 < v < T…

A: Given that, F¯=y,-x,z5 and ru,v=u cos v, u sin v, v where 0≤u≤3 and 0≤v≤π. Find ru×rv as follows.…

Q: Show that the vector v = [a, b, c]* is perpendicular to the plane P with equation ax + by + cz + d =…

A:

Q: In the figure, a vector a→ with a magnitude of 24.1 m is directed at angle θ = 68o counterclockwise…

A: Given that: Magnitude of a→ =24.1mAngle of a→ w.r.t. coordinate axes: θ=68°Angle of primed…

Q: Find the sine of the angle between vectors u = (-3, 2,6) and v = (3,2,6) by using the cross-product.

A: Given: The vectors u =(-3, 2, 6) and v=(3,2,6) Find : The sine of the angle between the vectors u…

Q: The vector equation r(u,v)=ucosvi+usinvj+vk, 0≤v≤4π, 0≤u≤1, describes a helicoid (spiral ramp). What…

A:

Q: Find the equation for the plane through the point Po = (3,4,3) and normal to the vector n=21 + 7j +…

A:

Q: Use the dot product to find the angle between the vectors v =i – 2j + k and w = i+j-2k. For which…

A: First we have to find the angle between the vectors v and w where v=i-2j+k and w=i+j-2k. For this,…

Q: Find the vector that makes equal acute angles with the positive coordinate axes and has length 2. O…

A: An acute angle is an angle that measure is less than a right angle (90 degrees). The magnitude of a…

Q: Prove the stated property of distance between vectors. d(u, v) = 0 if and only if u = v

A:

Q: +C, vnere B 1s parallel to the vector (1, 2) C is perpendicular

A:

Q: Find the component of the force 5i+7j-k in the direction of the displacement PQ, where P(3,0,1) and…

A: The given force is

Q: Find the equation for the plane through the point Po = (2,2,7) and normal to the vector n= 8i + 3j +…

A: Equation of plane passing through point P(x0,y0,z0) and normal to vector n=ai+bj+ck is given by:…

Q: which vector is NOT orthogonal to (cos θ, sin θ, 1)? a) (-sin θ, cos θ, 0) b) (-cos θ, -sin θ, 1)…

A: To find which vector is not orthogonal to (cosθ,sinθ,1)

Q: 19. What is the radius vector of the point on the terminal side of an angle in standard position if…

A:

Q: Find the angle between the plane z=0 and the plane passing through the points (0, 0, 0), (3, 1, 0),…

A:

Q: If A=(rsinOcosð,-cos2OsinD, tan, Inr) at P(7,0.08 rad,4.66 rad), determine the vector component of A…

A: Given:

Q: The angle between the unit vectors a and b is 0. By expanding the scalar product, show that (3a…

A:

Q: Find the angle between the vectors u = cos( /6)i + sin( /6)j, v = cos( 3/4 )i + sin(3/4 )j     (a)…

A: u=cosπ6i+sinπ6j,  v=cos3π4i+sin3π4j Now,…

Q: W,X ,Y,and  Z have the position vectors 3i+3j-4k, 4i+j+k, 2i+3k and 3i+j-4k respectively.…

A: The given position vectors areW = 3 i^ + 3 j^ - 4 k^X = 4 i^ + j^ + k^Y = 2 i^ + 3 k^ Z = 3 i^ +  j^…

Q: Two vectors of magnitudes A=3 and B=3v2 are lying on the xy plane as shown in the figure. What is…

A:

Q: What is a vector that is normal to the plane 4x+6(y-100)+7(z-200)=20? O A. 4i+6j-20 k O B. 8i+ 12 j…

A: Given query is to find the normal vector.

Q: The position vectors of A and B with respect to the origin O are a and b respectively. M is the…

A:

Q: The angle between the vectors A = 2j -3j +1k and B = 4i -3j +1k, in term of cos@ is !!

A: We use the definition of angle between two vectors to find the angle between the vectors A and B.

Q: 7. For vectors a and b in R? prove Lagrange's identity: ||a x b|? = ||a|| ||b||? – (a · b)?

A: for vectors a and b in ℝ2 prove -lagrange's identity a×b2=a2b2-a·b2

Q: Express in cylindrical components: (a) the vector from C(3, 2, –7) to D(-1, –4,2); (b) a unit vector…

A:

Q: ir kümelerinden kaç tanesi R' için bir tabandır? how many of the given vector

A: A subest S of R2 is a subspace if and only if it contains the null vector whenever it contains two…

Q: The vector i +bj-k is perpendicular to the vector I +2j-3k. what is the value of b?.

A: We know that if two vectors are perpendicular to each other then there dot product is zero. Given,…

Q: The position vectors of A and B with respect to the origin O are a and b respectively. M is the…

A: Given that position vector of A and B with respect to the origin O are a and b respectively. M is…

Q: The unit vector normal to the plane 2x + 3y + 6z = 7 is

A: The given plane is 2x + 3y + 6z=7. The unit normal vector to fx,y is defined as n=∇F∇F. The normal…

Q: Identify the surface with the given vector equation. r(0,0) = 2sinocosei +2sinosinej +2/1+sin?ok a.…

A: The minimum number of parameters needed to represent a surface is two. Therefore, in general, the…

Q: Give the unit vector in spherical coordinates that extends from M(5,1,2) to N(2,4,6).

A:

Q: The angle between the unit vectors a and b is 0. By expanding the scalar product, show that (3a - b)…

A:

Q: Evaluate F. dS, where F = and S is the helicoid with vector equation r(u, v) , 0 < u < 3, 0 < v <…

A: Given the vector field F→=y,-x,z2 and the surface S is parametrized as ru,v=ucosv,usinv,v,…

Q: Evaluate 1+ x² + y*dS, where S is the helicoid with vector equation r(u, v) 0 < u < 2, 0 < v < 2T

A: Given ∬S1+x2+y2dS where S is the helicoid with ru,v=u cos v, usin v, v, 0≤u≤2, 0≤v≤2π. Since ru,v=u…

Q: compute the dot products i ⋅ j, i ⋅ k, and j ⋅ k for the unit coordinate vectors. What do you…

A: Given: Compute the dot products i ⋅ j, i ⋅ k, and j ⋅ k for the unit coordinate vectors.

Q: Is the vector projection of B onto A, perpendicular to B? Is the angle between projAB and the vector…

A:

Q: The part of the plane with vector equation r(u, v) = (u + v, 2 – 3u, 1 + u – v) that is given by 0 <…

A: This is a problem of surface integral.

Q: 1. Determine the vector component of A = 3a, + 2ag – 6as tangential to the cone. %3D

A: The vector is A→=3ar+2aθ-6aϕ. Generally, the vectors are shown as sketched below:   Instead of…