Let ri = xi+y1j+2ık, r2 = x2i+ Y2j + zzk and r3 = x3i+ Y3j+ z3k %3D be the position vectors of P points P¡(x1, Y1, 21), P2(x2, Y2, 2) P1 and P3(x3, Y3, 23). Find an equa- Ps Ts tion for the plane passing through P1, P2 and P3. See fig- ure on the right

Q: Find a linear mapping G that maps [0, 1]×[0, 1] to the parallelogram in the xy-plane spanned by the…

A: A linear mapping G that maps [0, 1]×[0, 1] to the parallelogram in the xy-plane spanned by the…

Q: The vectors u=(1,-2,1), v=(3,0,-2) and w-(5,-4,0) are: O not coplanar coplanar

A: We can check whether the three vectors are co planar  or not as below

Q: Find a unit vector normal to the plane containing v=2i+3j-k and . w = -2i - 4j - 3k.

A: Given: v=2i+3j-k  w = -2i - 4j - 3k.

Q: Find a nonzero vector orthogonal to the plane through points P(2,1,5), Q(-1,2,3), and R(0,3,3), and…

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Q: Find a vector parallel to the line of intersection of the planes 2x + 3y – z =11 and 4x + 2y – 3z =…

A: We need to find a vector parallel to the line of intersection of the planes 2x+3y-z=11 and 4x + 2y -…

Q: 2. The plane containing the point P(Xg» Yor Zo) and having the normal vector n = (a, b, c) is…

A: Equation of a plane passing through the point Px0, y0, z0 and perpendicular to the vector n=a, b, c…

Q: Find a nonzero vector parallel to the line of intersection of the two planes – (2x + 3y + z) = -2…

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Q: Consider the vectors a = (-1,–2, 3), b = (3,0, –2), v = (2, –5, –4) and the plane P given by 2x + y…

A: The given vector is; a→=-1,-2,3b→=3,0,-2v→=⟨2,-5,-4⟩and Plane P is;2x+y=z+13⇒2x+y-z=13…

Q: Find a vector of magnitude 2 parallel to the line of intersection of the planes x + 2y + z - 1 = 0…

A: Find a vector of magnitude 2 parallel to the line of intersection of the planes x + 2y + z - 1 = 0…

Q: Find volume V(S) of the parallelepiped S in R3 determined by the (3,4,1); v3 following vectors: vị =…

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Q: Find a linear mapping G that maps [0, 1] x [0, 1] to the parallelogram in the xy-plane spanned by…

A: Consider the given figure with the following conditions:

Q: Find two unit vectors that are parallel to the yz-plane and are orthogonal to the vector 10i – 2j +…

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Q: What is equation of the plane passing through the point A(-2, –1, –3) and parallel to the vectors ā…

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Q: Let r = ri+y1j+z¡k, r2 = 1zi+ Y2j + zzk and r3 = r3i+ yaj + z3k be the position vectors of points…

A:

Q: Find a vector function that represents the curve of intersection of two surfaces: z = x² - y? and x²…

A: Idea  1) Solve the equation in term of only one variable (2) assume in term of parametric function…

Q: Let the solid form in the orthogonal coordinate system xyz be surrounded by points A, B, C, D and F,…

A: The given co-ordinates of vertices are A = (-4, 1, 3), B = (6, 1, 1), C = ( 9, 5, 0), D = (4, 5, 1),…

Q: Let u =(0,1,2) and v=(-2,2,0). Find %3D a. A vector w orthogonal to both u and v. b. The area of the…

A: The orthogonal vector both u and v is u×v . And the area of parallelogram is |u×v|

Q: The vector a in 3-space of length 3 lying in the yz-plane pointing upward at an angle of 5π/6…

A: let w= aiÌ‚ + bjÌ‚ + ckÌ‚   |w|= a2 + b2 + c2 = 1   W is lying in YZ plane and at an angle of 5π/6…

Q: Let u = 2i +j- 3k, v = i -j- k. %3D Find the vector projection Projuv o (2, 1, -3) O (1, 1, –3) O…

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Q: Find basis for the set of vectors in R'on the line y = nx. Where n=9.

A: Note that ℝ2 forms a vector space over the field of Real numbers ℝ. Let W =x1x2∈ℝ2  : x2 =9 x1 ,…

Q: Let ri = xịi+yıj+z¡k, r, = 1,i+ Yzj+ zzk and r3 = 1zi+ Y3j+ z3k be the position vectors of P1 points…

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Q: Let x 0 and y 0. (a) If xly, show that {x, y} is a linearly independ- ent set. (b) Extend the result…

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Q: Let ri = xji+yij+z¡k, r2 = x2i+ %3D %3D Y2j+ zzk and r3 = x3i+ Yzj + zzk be the position vectors of…

A: Vector equation of the plane passing through the points with position vector P1, P2 and P3  is given…

Q: Let v =5j and let u be a vector with length 3 that starts at the origin and rotates in the xy-plane.…

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Q: Write the equation of the plane with normal vector n = 4k passing through the point (7, 2, 5) in the…

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Q: Find x such that the vectors xi+2j -3k, i+j ,j+3k are Coplanars (lies the Same Plane) . on

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Q: Let ri = xji+yij+z¡k, r2 = x2i+ Y2j + z2k and r3 = x3i+ Y3j + z3k be the position vectors of points…

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Q: Find the vector parametric equation of the line that passes through the points P(6,–2, –1) and…

A: The vector equation of a line is  . Where represent the position vector,  represent the direction…

Q: Describe the span ot the two vectors (1,0,9) and co. 1,15 geometricellyy in

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Q: Find the Directional Derivabive of Pary,2)= x-yz'+ at the point CI, -2, 0) in the direcbion of…

A: A scalar function is a rule to associate a number to a particular point in space. The directional…

Q: 1) in triple dot of vectors |(A x B).C| = |(C x A). B|, where the vectors %3D A,B, and C are not…

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Q: (a) The vector v in 2-space of length 5 pointing up at an angle of 37/4 measured from the positive…

A:

Q: Let u = -2i + 3j, v = 6i - j, w = -3i. Find the specified vector or scalar lu + vl2 - lu - vl2.

A: The given vectors are  u=-2i+3j , v=6i-j  ,w=-3i To find: the value of u+v2-u-v2. Solution: Since,…

Q: Let ri = xi+yıj+z¡k, r2 = x2i+ Y2j + z2k and r3 = x3i+ Y3j + z3k be the position vectors of points…

A:

Q: 4. Show that the vectors v1, v2, and v3 are linearly dependent, where Vị = V2 = and v3 =

A: If we have a set of vectors v1, v2, v3, ..., vn, then they are said to be linearly dependent if…

Q: Write the equation of the plane with normal vector n = 4k passing through the point (7, 2, 5) in the…

A: The equation of plane with normal vector (a,b,c) passing through the point p0 =(x0 ,y0 ,z0 ) is…

Q: 6. Let x#0 and y‡0. (a) If x 1 y, show that {x, y} is a linearly independ- ent set. (b) Extend the…

A:

Q: Find a nonzero vector u with initial point P(2,-3,6) such that: (a) u has the same direction as v =…

A: Find a nonzero vector u with initial point P(2,-3,6) such that: (a) u has the same direction as v =…

Q: a) Find the distance between two skew lines L1 and L2 such that L1: 1+ 4t, y=5-4t, z=-1+5t  L2:…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: Q3/ Show that the vectors (2i + 3j + 6k), (3i – 6j + 2k)and (6i + 2j + 3k) Are mutually…

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Q: Find the normal unit vector perpendicular on à and B for A = 2i+3j-k and B=-j+2k %3D

A: To find the normal unit vector perpendicular to A and B

Q: Exercise 6 Is the set of vectors V = {i² + 2t – 1,3t + 5, -t² + t + 6} linearly dependent?

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Q: Find x if the vectors a =i+j+ 2k;b = xi-j+k; coplanar or linearly dependent.

A: Given,       a=i+j+2kb=xi-j+kc=3i-2j-k

Q: Is it true or false that the vector u (3, 3, –3) is a normal vector to the plane -x – y + z + 3 = 0…

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Q: Find a vector that is orthogonal to the plane containing the points P = (3,0, 1), Q = (4, –2, 1) and…

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Q: Write the equation of the plane with normal vector n = (1, 3, 4) passing through the point (3, -1,…

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Q: -2 - are the vectors U, = %3D Ug = %3D in M22 are lineor ly depon dent linearly in dependet.

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Q: Find a nonzero vector parallel to the line of intersection of the two planes 4y – 3x +3z = -2 and z…

A: Answer

Q: Find a nonzero vector parallel to the line of intersection of the two planes 5z – (3x+3y) : 2 and 2x…

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.