Part I. Vector a'(0,1,0), b (1,0,1) calculate (2 points for each question, 10 points in total) (1)a+ b, (2) 3а , (3) The length of a', (4) The cross product a'x b, (5) Prove vector(-2,2,2) is perpendicular to a'x b

Q: Find both the vector equation and the parametric equations of the line through (0,0,4) in the…

A: Explanation of the answer is as follows

Q: Find an equation of the plane. The plane through the point (1, 7, 2) and with normal vector 4i + j –…

A: Given:   Point : ( 1 , 7 , 2)  Normal vector : 4i + j - k

Q: Find the coordinates of a point P on the line and a vector v parallel to the line. x − 3 _ y + 1 4 2…

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Q: Determine the distance between the largest and smallest value of the parameter t belonging to the…

A: Given vector function r(t)  =81-t2  ,  t2-4  , ln3t

Q: Find the scalar and vector projection of the vector b = (−1, −1, 1) onto the ve Scalar projection…

A: Scalar Projection :  The length of the projection is called scalar projection. Vector Projection :…

Q: Let(vector u)=(-2,-3), (vector v=(-3,-5) and vector w=(-2,-3). Find the vector x that satisfies…

A: u=(-2,-3)v=(-3,-5)w=(-2,-3)Calculating x:

Q: Find the equations of the normal line and the tangent plane to the parametric surface defined by the…

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Q: Fine the Vector function that represents the curve of intersection of the two surfaces. The cone z=…

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Q: let angle thetha be measured from x1 axis counterclockwise to a line drawn normal to the boundary…

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Q: Find parametric and vector equation for line segment which  reach (2, 2.4 , 3.5) point and parallel…

A: The parametric equation  of a line segment in three variables is given by the expression: Here, is…

Q: find vector v that is orthogonal to vector (0,1,0,0,0) Select one: a. v=( -1, 0, s, t, w) b. v=( r,…

A: Two vector are orthogonal if their dot product is zero.

Q: Let the temperature T in degrees at the point (x,y,z), with distances measured in cm, be T(x,y,z)=3x…

A: Here I am attaching image so that you understand each and every step.

Q: Find an equation of the plane through the point (4, 5, 1) and with normal vector 6i + j - k. Show…

A: The plane passing through a point A(a) with the normal vector n is given by: r·n=a·n where,…

Q: Find a vector parallel to the line of intersection of the planes 5x−y+2z=4 and 4x−y−3z=0v→=  (b)…

A: Find a vector parallel to the line of intersection of the planes 5x−y+2z=4 and 4x−y−3z=0

Q: . Value of t such that vectors a=, b=, c= are coplanar.

A: Value of t such that vectors a=<1, 1, 1>, b=<2, -4, 0>, c=<1, t, 3> are coplanar.

Q: 1 Find equations in vector and rectangular form for the a) osculating plane, b) normal plane, and c)…

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Q: sketch the space curve represented by the intersection of the surfaces. Then represent the curve by…

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Q: Find the directional derivative of f (x, y, z) = 2z²x + y³ at the point (2, 2, 1) in the direction…

A: The solution is given as

Q: 1. Determine the vector and parametric equations of the plane containing the point (1, -5, 9) and…

A: Equation of plane

Q: Find the directional derivative of P(x, y, z) = x ^ 3 - x * y ^ 2 - z at B(1,1,0) in the direction…

A: The given function is Px, y, z=x3-xy2-z We have to find the directional derivative of Px, y, z at…

Q: example of a vector equation for a line in 3 dimensions which doesn’t intersect the plane at z = 0

A: Given that:a vector equation for a line in 3 dimensions which doesn’t intersect the plane at z = 0

Q: 11. It is the distance of the point A on the terminal side of angle in standard position from the…

A: Option C

Q: Talkin Consider in an orthonormal system (0; 1,j), the straight line (D) of equation: 2x-y+4 0, and…

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Q: What do you mean by vector product/cross product?

A: We have describe vector product/cross product.

Q: How does translation T along vector AB affect the points in the plane?

A: Some information in the question is missing.

Q: Find two unit vectors that are normal to the plane determined by the points A (0, -1, 2),B…

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Q: Write the equation of the plane with normal vector n passing through point Po in the scalar form ax…

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Q: Find a vector equation and parametric equations for tha line sagmant that joins P to o. A-6, 8, 0).…

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Q: Find the equation of the plane with normal vector n = (8, 7, 3) passing through the point (4,,9) in…

A: Use equation of plane and put value

Q: Find an equation of the plane that passes through the point Po(0,2, – 2) with a normal vector n =…

A: Find an equation of the plane that passes through the point p0 (0,2,-2)with a normal vector…

Q: Given two points P(-2,1,–3) and Q(3,–5,4) evaluate PQ. (a) į-4j+k (b) -į+4j-k (c) 5i – 6j+7k (d) -5į…

A: Given that P(-2,1,-3) and Q(3,-5,4) evalute PQ vector.     Here i attached solution of this…

Q: A plane passes through the points A, B and C with coordinates (9,1, –4), (13, —4, 7) and (4, 2, −5)…

A: For the given points on the plane the cross product AB→ ×AC → = -6i-51j-21k or can be written as…

Q: Given two vectors u = (2,-1,3) and y=(-3,2,4). Calculate vxu. (a) (1,-1,–7) (b) (-1,1,7) (c)…

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Q: of the vector 5 . ) Petermine line an dlependence X,= (1,0,2,1) X2 (3,1,2, D (4,6, 2,4) %3D X3 X4 =…

A: Vectors are linearly independent if there exist non zero value of ci's such that:…

Q: Determine the scalar equation of a plane through the point (3, 0, 4) and having normal vector n= (2,…

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Q: Q5: Plot the function y= 2 tan(t)-cos(mt) labeling the x and y axes. Where t is a vector containing…

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Q: Let u be the vector with initial point P(5, -7) and terminal point Q(-8, 9). (a) Graph u in the…

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Q: find vector v that is orthogonal to vector (0,1,0,0,0) Select one: a. v=( r, 0, s, -t, w) b. v=( r,…

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Q: 5. Suppose that we have a function f(x, y) and a point P = (a, b) in the domain where ) and = 0.…

A: Given that for the multivariable function f(x,y) , ∂f∂x=0 & ∂f∂y=0  at P=(a,b).

Q: Let the space curve be the intersection of two surfaces z = x² + y² and y² + z² = 1. Draw this curve…

A: As per the question we are given two 3d surfaces defined by the equations : z = x2 + y2   ;   y2 +…

Q: 1. through the points P(1, -3, 2) and Q(9, 2, 0) [vector equation] 2. with direction vector d (5, 3,…

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Q: Cross-product of vectors   ⟨2,−1,3⟩×2⟨4,2,1⟩=?

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Q: A vector orthogonal to the plane through the points P, Q and R Calculate the area of the triangle…

A: a)  The given vertices are P(-4,-4,-4), Q(0,5,-1) and R(3,1,2) PQ =0+4,5+4,-1+4 =4,9,3PR  = 3+4,…

Q: Let u be the vector vith initial point P(5, -3) and terminal point Q(-1, 4). (a) Graph u in the…

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Q: 4. Find an equation for the plane through (1, 3, 4), with normal vector .

A: topic- equation of plane with point and normal vector

Q: Determine an equation (in variables x, y, z) for the plane through the point (8, 1, 4), and with…

A: GIVEN- EQUATION OF PLANE PASSING THROUGH THE POINT (8,1,4) AND WITH VECTOR (-2,6,-2) TO FIND-…

Q: (Ъ) (aU) · (bV) = ab(U.V)

A: Explanation of the answer is as follows

Q: 1. The cross product of the vectors u = (a,, az, az) and v (bı, b, b;) is the vector ij k| u X v =…

A: given,         u=a1,a2,a3 and v=b1,b2,b3   and u=1,0,1 and v=2,3,0