The position vectors of the non-collinear points A, B, C are a, b, c respectively. E lies on BC with |BE|/|EC| = 2/3, F lies on Ca with |CF|/|FA| = 1/4, and G lies on AB produced with |GB|/|GA| = 1/6. Determine the position vectors of E, F, G in terms of a, b, c and deduce that E, F, G lie on a straight line. Vectors

Q: Given the points P, Q, and R, how do we know that the vectors PQ (cross product) PR is perpendicular…

A: Given that there are three pints in the plain P,Q and R.The cross product between PQ and PR is…

Q: Let A and B be two points with position vectors A and B, respectively. Show that the line passing…

A: To derive the vector form of the required straight line

Q: Given the vectors u = (-3, 1, 2), v = (1, 2, 1), and w = u + mv, determine the value of m so that…

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Q: Let A and B be nonzero vectors. Find k e R such that A – kB is orthogonal to A + B.

A: We know if a vector P→ is orthogonal to another vector Q→ then scalar product (dot product) is zero.…

Q: .If the position vectors of the points A, B, C, D are a = i+j+k, b 2i + 3j, c= 3i+ 5j - 2k, d k-j.…

A: Given:  a=i+j+k, b=2i+3j, c=3i+5j-2k, d=k-j We know that if A=x1i+x2j+x3k, B=y1i+y2j+y3k then…

Q: The position vectors of points A, B and C are OÃ = 2i – j + nk , OB =ỉ+3j – 2K and OC = -i – mỉ – 3k…

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Q: Given the points M[0.1, -0.2, -0. 1), N(-0.2, 0.1, 0.3), and P(0.4, 0, 0.1), find: (a) the vector…

A: This question can be solved using the concept of finding the vector joining two points and dot…

Q: For what value of t, if any, is the vector (8t, 2) parallel to the vector (4, -1) ?

A: Given vectors:A= <8t, 2> and  B= <4, -1>

Q: find "a" so that the vectors v=i-ja and w=2i+3j are orthaganal?

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Q: The vector from the origin to point A is given as 6a, – 2a, - 4a., and the unit vector directed from…

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Q: Find the given vector : Vector of length 3 in the direction of v = 4i + 3j

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Q: The lines L₁ and L2 meet at the point P. The line L3 is coplanar with L₁ and L2 and is perpendicular…

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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: Find vector u with the magnitude given that satisfies the given conditions. v (7, -1, 3), ||u || -…

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Q: Given a vector A = 3a, - 2a, - 16a, find its components normal to the plane 2x + 2y – z + 8 = 0 and…

A: We have to determine component normal to the plane

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 vector whose magnitude is 5 and which is in the direction of the vector 4i- 3j + k. Q.2

A: Since you have asked multiple question As per our policy, we will solve the first question for you.…

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: The vector u has length 15 and points at an angle of 25° The vector v has length 10 and points at an…

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Q: ABC is a triangle and P, Q are the midpoints of AB, AC respectively. If AB = 2x and AC = 2y, express…

A: Given P,Q are the mid-points of 2 sides AB,AC of the triangle ABC. Also, AB=2x, AC=2y.

Q: The vectors T and N, when defined, (a) are parallel. (b) are orthogonal. (c) have a zero cross…

A: T is the tangent vector and N is the normal vector.  So they are always orthogonal to each other.

Q: The vector u has length 10 and points at an angle of 15∘ The vector v has length 18 and points at an…

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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: The vector from the origin to point A is given as 6a, – 2a, – 4a., and the unit vector directed from…

A: The vector from the origin to point A is 6ax-2ay-4az Therefore, vector A is A→=6,-2,-4 Now, the unit…

Q: 2. Let each of the vector A = -8a, + 2ay – 4a,, B = 3ax – 4ay |3D Z> + 5a,, and C = 4ax + 5a, extend…

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Q: For what value of t if any, is the vector 5t, 1] parallel to the vector (4, -1) ?

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Q: The vector u has length 14 and points at an angle of 20° The vector v has length 16 and points at an…

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Q: At what value of k are these vectors collinear?

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Q: In each part, find the two unit vectors in R² that satisfy the given conditions. 1. The two unit…

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Q: Given the points X(1, -2, 1) and Y(0, 3, -2), find a vector that is collinear with XY and has a…

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Q: Consider the graph provided find the length of the vectors u and i

A: Solution and Explanation is given below....

Q: 2. Given the points P(-1,3, 1), Q(0,5, 2), R(4,3, –1), (a) Find a nonzero vector orthogonal to the…

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

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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 and b are vectors with the same initial point and angle between them, prove that||a - b||2 =…

A: Consider the given expression as, a-b2

Q: i +j +k andi – j -k.

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

Q: Prove that the vectors u = (cos θ, −sin θ) and v = (sin θ, cos θ) are orthogonal unit vectors for…

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A: Given: X(1, -2, 1) and Y(0, 3, -2).

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Q: find the two-dimensional vector equation of the line r(t) = a +t*m such that - r(0) has norm 8 and…

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Q: 27-j+4k sī +2j-2k.

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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: Two vectors are given as A=-1a, + 6a, + 6a, B=-6a, + 5a, + 6a, Find a unit vector normal to the…

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Q: What value(s) of a are necessary for vectors i = (2, a), and w = (4a, 3) to span R?

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Q: Find a vector B for which that is parallel to A =-6i +3j –2k but has the opposite direction.

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Q: Find the vectors that satisfy the stated conditions. 23. (a) Oppositely directed to v (3,-4) and…

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Q: Use the points P(3, 1) and Q(7, 1) to find position vectors equal to PQ and QP.

A:

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.