6. Show that the points A, B and C with position vectors d = 32 – 4) – 4k, b = 2î – ĵ + k and & = î – 33 – 5k, respectively form the vertices of a right-angled triangle.

Q: If v = 2i - 5j and w = i - j, find the vector projection of v onto w.

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Q: и Match the vectors p, q, r, s, and t with one of the combinations of the u and v vectors from the…

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Q: B) Given the vectors u = 7i+ 2j -k, v = 2i-3j-6k. Find the projection of %3D u onto v.

A: Using projection vector formula.

Q: bäi The cross product of the vectors 3ax + 4ay – 5az and -ax + ay - 2az * ,is -3ax+11ay-7az O -3ax +…

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Q: Find the projection of d on the plane P if a 4 and P has the equation x + y+ z = 1.

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Q: If a line is parallel to the vector 2i + 9j + 8k and passes through the point P = (6, 8, 8), then…

A:

Q: Consider the hyperboliode H defined by the equation:  x^2+y^2−z^2=1 and the point P0=(1,4,4) in H.…

A: We are given the following hyperboloid : x2 + y2 - z2 = 1 And we have to choose the correct…

Q: #1. The area of the parallelogram spanned by the vectors v = (A) – 22 (B) 22 (C) 23 (D) 8 (E) 2 (5,…

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Q: If a line is parallel to the vector 2i + 7j + 10k and passes through the point P = (6, 10, 5), then…

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Q: Consider the vectors x1 = (8, 6)T and x2 = (4, −1)T in R2. Determine the length of each vector.

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Q: IF A =2âx+4ây , B =-3âx+âz and C =2ây+3âz, the vector in the plane containing A and C and…

A: Given: Let us consider the given values as ,  A→,B→,C→A→=2a^x+4a^yB→=3a^x+a^zC→=2a^y+3a^z

Q: 1. Find the volume of the parallelepiped PQRS if the vectors V(PQ), V(PR), and V(PS) are…

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

Q: The vector parallel to intersection line of the two planes 4x-y+z=2 and 2x+y-2z=3 is: O0 -3i+6j-3k…

A: Two planes are given: 4x-y+z=2 and 2x+y-2z=3

Q: Suppose vectors v, w, and z sum to the zero vector. If v = 8.2i - 3.8j and w = 4.1i + 1.3j, find z.

A: We have to solve

Q: 10- With respect to the origin O, the vertices of a triangle ABC have position vectors OÁ = 2i + 5k,…

A: (a) Given that position Vectors, OA→=2i+5kOB→=3i+2j+3kOC→=i+j+k   As we know that, two vectors are…

Q: 1. Show that the necessary and sufficient condition for two nonzero vectors A and B to be…

A: As per the company guidelines We can first question only. Please understand. Kindly repost the…

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: 4. Given the two vectors rA =-ax - 3ay - 4az and re= 2ax + 2ay + 2 az and point C (1, 2, 3) find: a)…

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Q: ±). 1.4 Use vectors to decide whether the triangle with vertices P(1,–3,–2), Q(2,0, –4) and R(6, –2,…

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

Q: The vectors A 6i + 4j – 8k and B = xi + 6j + zk are known to be parallel. Find x and z x = 9. -12

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Q: Find the value of k for which the vectors 2 1 2 and -3 5 k are orthogonal. k =

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Q: 24. Find v x w, where v and w are vectors of length 3 in the xz-plane, oriented as in Figure 16, and…

A: The vectors v and w are in the x-z plane. The cross product of both vectors will yield a vector…

Q: For the points A(1, 5, -7) and B(3, 0, 1) vector AB has coordinates

A: Given point A and B. Find vector AB.

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

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Q: If the points with position vectors 10i+3j.12i-5j and ai+11j are collinear, Find the value of a.

A: The position vectors of the points which are collinear is: 10i+3j ,     12i-5j ,      ai+11j

Q: Show that if v and w lie in the yz-plane, then v × w is a multipleof i.

A: Both vectors v and w lie in the y-z plane, which means they must have zero magnitude component along…

Q: Given the vectors P=3i - j +k, Q =4i +Qy j-2k, and S=2i -2j +2k,determine the value of Qy for which…

A: given the three vectors P=3i-+j+k , Q=4i+Qyj-2k  and S=2i-2j+2k are coplanar vectors. we need to…

Q: Find a set of vectors {u, v} in Rª that spans the solution set of the equations { y + 4z 0, - 5w +…

A:

Q: 22. Given that the vectors a = 3i-4j-4k, b = 2i-j + 3k, and c = 4i-3j +5k. The %3D %3D projection…

A: Here let d= a+b d= a + b⇒d = 5 i -5j -kProjection length of vector d = (a+b) on c is given by |d|…

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: If a line is parallel to the vector 6i + 7j+ 2k and passes through the point P= (5, 1,7), then the…

A: Given that a line is parallel to the vector 6i+7j+2k and passes through the point P=5, 1, 7. We know…

Q: If a line is parallel to the vector 4i + 1j + 4k and passes through the point P = (3, 7,9), then the…

A:

Q: Given the vectors A = 3i – 5j + 2k and B = 7i + j – 2k. Find the vector projection of A onto B.

A:

Q: 16. Consider this regular octagon. If AB =d and BC =v, express each %3D vector in terms of u and v.…

A: Considering OD on x-axis. Angle at center O made by any side of regular octagon is 45degrees.

Q: If a = i+3j – k, b = 2i + 4j – 2k and c = -i + 2j +4k, find a number A such that d = a +b+ Xc is…

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Q: If the vectors b, c, d are not coplanar, then can we say that the vector (ā×b)×(ē×d) + (ā×ē) × (dxb)…

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Q: If a line is parallel to the vector 3i + 9j+ 6k and passes through the point P = (5, 5, 4), then the…

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Q: The vectors v = (3, 4) is projected onto the x-axis The length of the projection is %3D A/

A: We have to find

Q: Find an equation for the plane that passes through the point (1, 2, 3) parallel to u = 2i + 3j + k…

A: Given:u = 2i + 3j + k and v = i - j + 2kFirst find cross product

Q: 10- With respect to the origin O, the vertices of a triangle ABC have position vectors OA = 2i + 5k,…

A: (a) To show that triangle ABC is a right angled triangle, use scalar product of vector, that is…

Q: Consider the points K (−1, 0, 7) and M(1, −2, 5), the vector MN = (1, 4, −3), and the line  I with…

A: We have to solve given problems:

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

A:

Q: Find the angles that vector D 2.5 - 8.5 + k) m makes with the x-, y-, and z- axes.

A: Follow the procedure given below.

Q: Consider the vectors x1 = (2, 1)T and x2 = (6, 3)T . Determine the length of each vector.

A: Given: x1=2i+j, x2=6i+3j we know that for any given vector v=ai+bj, length l of vector is given by…

Q: 2 -a) Find a unit vector perpendicular to the plane that passes through the points P(1,4, 6), Q =…

A:

Q: 7. If a -i+3j – k, b = 2i + 4j– 2k and c = d = a+b+ de is parallel to the yz-plane. -i+ 2j + 4k,…

A: given a=i+3j-kb=2i+4j-2kc=-i+2j+4k

Q: (3) Write the equation of the plane which is perpendicular to the vector ñ² = (2₁-1₁-2) and touches…

A: To find- Write the equation of the plane which is perpendicular to the vector n→ = 2, -1, -2 and…

Q: O Are the three vectors: V= 4i-8j+kV= 2i+j-2k and ū;=3i-4j+12K in the Same Plane?

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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: For what values of y are the vectors yi – 2j + 3y k and 6i +j+ 3y k orthogonal.

A: To find the values of y for which the vectors yi-2j+3y k and 6i+j+3y k are orthogonal.

Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.