W,X ,Y,and  Z have the position vectors 3i+3j-4k, 4i+j+k, 2i+3k and 3i+j-4k respectively.     Find the angle between the line WY and the plane XYZ.

Q: The unit vector which is perpendicular to both r1=8ax+3ay- 2az , r2=2ax+7ay-6az is equal to

A: Given, vectors arer1=8ax+3ay-2az, r2=2ax+7ay-6azThe unit vector perpendicular to both vectorsr1, r2…

Q: Show that the lines L1 : x = 4 – t, y = 2t, z = 2+t and L2 : x = 1+ 4t, y = 3 – 8t, z = 5 – 4t are…

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Q: The vectors A 5i + 7j – 8k and B 3i + yj + zk are known to be parallel. Find y and 2 y = = Z

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Q: Is the line x = 6- 3t, y = 8 + 8t, z= -t parallel to the plane 3x + y -z=2? Give reasons for your…

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Q: Find the vector equation for the line of intersection of the planes 5x +y-3z= -3 and 5x + z = -1 r =…

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Q: find the acute angle between the given linesby using vectors parallel to the lines. 2 - x + 2y = 0,…

A: 2-x+2y = 0⇒2y = x-2⇒y = x-22⇒y = x2-1             --(1)So, vector parallel to (1) is a→ = 12i ,…

Q: find the acute angle between the given linesby using vectors parallel to the lines. y = x, y = 2x +…

A: Here the acute angle between two lines is to be calculated.

Q: Find the vector equation for the line of intersection of the planes x + 4y + z = -5 and x + 4z = 0…

A: Here we have to find the vector equation for the line of intersection of the planes x+4y+z = -5 and…

Q: Find the coordinates of the point P on the line and a vector v parallel to the line. X = 4t, y = 4 –…

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Q: Consider the planes x – 10y – 2z = 3 and 2x – y + z = -13. Find the following. 1. A vector…

A: The given planes are: x-10y-2z=3 and 2x-y+z=-13 Let us assume z=t on the line of intersection, where…

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

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Q: Show that the lines L1 : x =1– t, y = 2t, z = 4+t and L2 : x = 1+ 3t, y= 3 – 6t, z = 5 – 3t are…

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Q: Consider the points P(2, −1, 0), Q(5, 0, 0), and R(3, 1, √5). Find the coordinates of the vector QS…

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Q: Given A = x(2z -y) + ŷ(3x-2z) – 2(x+3y), find a unit vector parallel to A at point P =(-2, 1,–3).…

A: Vector valued functions are the rule that used to fix or associate a vector to a particular point…

Q: Find the angle between the vectors u = 2i +j+9k, v = 6i – 12j and sketch the vectors and the angle…

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Q: 14. The accompanying figure shows nonzero vectors v, w, and z, with z orthogonal to the line L, and…

A: Given V and W make any equal angle with L. |v| = |w| Z and L are orthogonal to each other.

Q: Find the angle between the vectors u = /2i - 9j and v =v2i+j-2k.

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Q: What angle does the line of intersection of the planes (x 2y + 3z 5 and 4x+y-2z-7 make with the…

A: The dot product of two vectors u→, v→ is calculated using the formula: u→·v→=u→v→cosθ, where θ is…

Q: Consider u, v, w € R'as u = 1 1 w = 2 Choose the vectors that have the HIGHEST cosine similarity. u…

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Q: Find the coordinates of the point P on the line and a vector v parallel to the line. x = 5t, y = 9 –…

A: Let's find.

Q: Let the cosine of the angle between the vectors p and q be à such that 2p + q = i + j and p + 2q =…

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Q: Find the values of x such that the angle between the vectors u = (2, 1, -1) and v = (1, x, 0) is…

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Q: 16. The accompanying figure shows nonzero vectors v, w, and z, with z orthogonal to the line L, and…

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Q: Given v = 2i + j, w = 6i + 3j, find projw v. Then decompose v into two vectors, v1 and v2, where v1…

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Q: Find the coordinates of the point P on the line and a vector v parallel to the line. x = 7t, y = 4 -…

A: Given, The equation of the line x = 7t, y = 4-t, z = 3+9t We have to find the coordinate of the…

Q: Find the angle between the two vectors u 2i+ j-7k and v =i-j+ 4k

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Q: Find the equation of the plane passes through the point P(4,2,1) and perpendicular to the vector…

A: Equation of the plane passing through point P(x0,y0,z0) on the plane and perpendicular to the vector…

Q: The unit vector which is perpendicular to both r1=8ax+3ay-2az, * r2=2ax+7ay-6az is equal to a)…

A: Given That: The unit vector is perpendicular to r1=8ax+3ay-2az & r2=2ax+7ay-6az

Q: A=\+2) -Kg B=-i+j+ K, Csitk, D= i+2j-K which vectors are: @ perpendicular parallel-

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Q: Find the vector equation for the line of intersection of the planes 5x + 4y + 5z = 1 and 5z + z = -4…

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Q: Consider the planes given by the equations x + 2y – 2z = 1, 4.x – y + 2z = 7. (a) Find a vector i…

A: Explanation of the solution is given below...

Q: Consider the planes given by the equations x−3y−2z=1, 4x−y+3z=8      Find a vector ' v ' parallel…

A: We have given the equation of plane-                      x−3y−2z=1,                     4x−y+3z=8.…

Q: Find the coordinates of the point P on the line and a vector v parallel to the line. X = 6t, y = 8 –…

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Q: Find the coordinates of the point P on the line and a vector v parallel to the line x = 3t, y = 1 –…

A: Given parametric equation of line x=3t, y=1-t, z=3+5t

Q: For two nonzero vectors u and v, if u v=0 then u and v are parallel.

A: Given that u and v are two nonzero vectors   Dot product of two nonzero vectors u.v=|u|.|v|…

Q: If u, and u2 are orthogonal unit vectors, and v = au, + bu2, find v•uq. = 'n.a v•u,

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Q: The plane that contains the point (3, –1, –4) and the line with equation = y – 1 = 3z is orthogonal…

A: Let:x-02=y-11=z-013=kL: x=2k, y=k+1,  z=k3P1:(3, -1, -4)Put k=0, k=1P2: (0, 1, 0)P3: (2, 2,…

Q: Given v = 3i - 2j, w = 2i + j, find projw v. Then decompose v into two vectors, v1 and v2, where v1…

A: Given that, The vectors  v = 3i - 2j, w = 2i + j. We have to find projw v By using a formula, projw…

Q: A line L has a direction vector v = 2i t 4j+ v5k and passes through the Find the distance between…

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Q: Given the planes 3x – 6y – 2z = 15 and 2x + y – 2z = 5 anser the following. (a) Find a vector…

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Q: Find the vector equation for the line of intersection of the planes x – 5y + 2z = 1 and æ + z = 1 |…

A: To find the vector equation of the line of intersection of given planes.

Q: Find the vector and the cartesian equations of the plane containing the point î+ 2j – k and parallel…

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Q: Consider the lines 3 - z x – 5 l1 : 2 and I. Find two unit vectors that are parallel to l1.

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Q: Find the angle between the vectors u and v to a nearest degree: u = j + k, v = i + 2j – 3k

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Q: Find the acute angle between the two vectors A = 2a, +a, + 3a, and B = ax – 3ay + 2a, by using the…

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Q: perpendicular to the plane formed by the vectors w1 = [1 2 1] and w2 = [2 2 3], with the z axis

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Q: Which of the following planes are parallel to the vector (4, 1,-5)? Select one or more: O a. 6x + y…

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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.