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Chapter 1: Problem 14

Given that \(\mathbf{A}+\mathbf{B}+\mathbf{C}=0\), where the three vectors represent line segments and extend from a common origin, must the three vectors be coplanar? If \(\mathbf{A}+\mathbf{B}+\mathbf{C}+\mathbf{D}=0\), are the four vectors coplanar?

Short Answer

Expert verified
What about vectors A, B, C, and D if A + B + C + D = 0? Answer: To determine if vectors A, B, and C are coplanar, we can compute the scalar triple product (AxB).C. If the scalar triple product equals 0, the vectors are coplanar. For the case of A + B + C + D = 0, we can similarly compute the scalar triple product of A, B, and D. If the scalar triple product equals 0, the vectors A, B, C, and D are coplanar.

Step by step solution

01

Define coplanar vectors

Vectors are said to be coplanar if they lie in the same plane. In order to determine if the given vectors are coplanar, we can use the concept of scalar triple product. The scalar triple product gives the volume of the parallelepiped formed by the three vectors. If the volume is zero, the vectors are coplanar since they don't form a 3D object; otherwise, they are not.
02

Use the scalar triple product formula

The scalar triple product formula can be defined as the dot product of one vector with the cross product of the other two vectors: (AxB).C
03

Compute the scalar triple product for A, B, and C

Using the formula from step 2: scalar_triple_product_abc = (AxB).C
04

Check for coplanarity of A, B, and C

If scalar_triple_product_abc equals 0, then A, B, and C are coplanar: coplanar_abc = (scalar_triple_product_abc == 0)
05

Compute the scalar triple product for A, B, and D

We know that A + B + C + D= 0, which means D = - (A + B + C). Now, find the scalar triple product of A, B, and D: scalar_triple_product_abd = (AxB).D
06

Check for coplanarity of A, B, C, and D

If scalar_triple_product_abd equals 0, then A, B, and D are coplanar: coplanar_abcd = (scalar_triple_product_abd == 0)
07

Conclusion

Based on the steps above, if coplanar_abc and coplanar_abcd are true, this means the three and four vectors are coplanar, respectively.

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Most popular questions from this chapter

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 \(\mathbf{R}_{M N} ;(b)\) the dot product \(\mathbf{R}_{M N} \cdot \mathbf{R}_{M P} ;(c)\) the scalar projection of \(\mathbf{R}_{M N}\) on \(\mathbf{R}_{M P} ;(d)\) the angle between \(\mathbf{R}_{M N}\) and \(\mathbf{R}_{M P}\).

Vector A extends from the origin to \((1,2,3)\), and vector \(\mathbf{B}\) extends from the origin to \((2,3,-2)\). Find \((a)\) the unit vector in the direction of \((\mathbf{A}-\mathbf{B})\); (b) the unit vector in the direction of the line extending from the origin to the midpoint of the line joining the ends of \(\mathbf{A}\) and \(\mathbf{B}\).

Given the vectors \(\mathbf{M}=-10 \mathbf{a}_{x}+4 \mathbf{a}_{y}-8 \mathbf{a}_{z}\) and \(\mathbf{N}=8 \mathbf{a}_{x}+7 \mathbf{a}_{y}-2 \mathbf{a}_{z}\), find: ( \(a\) ) a unit vector in the direction of \(-\mathbf{M}+2 \mathbf{N} ;(b)\) the magnitude of \(5 \mathbf{a}_{x}+\) \(\mathbf{N}-3 \mathbf{M} ;(c)|\mathbf{M}||2 \mathbf{N}|(\mathbf{M}+\mathbf{N})\)

Given the vector field \(\mathbf{E}=4 z y^{2} \cos 2 x \mathbf{a}_{x}+2 z y \sin 2 x \mathbf{a}_{y}+y^{2} \sin 2 x \mathbf{a}_{z}\) for the region \(|x|,|y|\), and \(|z|\) less than 2, find \((a)\) the surfaces on which \(E_{y}=0 ;(b)\) the region in which \(E_{y}=E_{z} ;(c)\) the region in which \(\mathbf{E}=0 .\)

A circle, centered at the origin with a radius of 2 units, lies in the \(x y\) plane. Determine the unit vector in rectangular components that lies in the \(x y\) plane, is tangent to the circle at \((-\sqrt{3}, 1,0)\), and is in the general direction of increasing values of \(y\).
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