The vectors a⃗ and b⃗ both lie in the x z plane. The vector c⃗ is equal to a⃗ ×b⃗. Which of thes

step 1 So given A plus B plus C is equal to 0, then A, B, C are in front plane and as respectively by t

The vectors a⃗ and b⃗ both lie in the x z plane. The vector...

The vectors $\vec{a}$ and $\vec{b}$ both lie in the $x z$ plane. The vector $\vec{c}$ is equal to $\vec{a} \times \vec{b}$. Which of these statements about $\vec{c}$ must be true? (1) The $x$ and $y$ components of $\vec{c}$ are zero. (2) The $y$ and $z$ components of $\vec{c}$ are zero. (3) The $z$ and $x$ components of $\vec{c}$ are zero.

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Key Concepts

Right-Hand Rule

The right-hand rule is a common convention used to determine the direction of the cross product vector. By pointing the index finger in the direction of the first vector and the middle finger in the direction of the second vector, the thumb will point in the direction of the cross product, ensuring consistency in three-dimensional space.

Normal Vectors to Planes

In vector algebra, a normal vector is one that is perpendicular to a particular surface or plane. When two vectors lie in a given plane, their cross product yields a normal vector to that plane, which is useful in applications like determining orientations and in physics for understanding forces.

Orthogonality in the Cross Product

A key property of the cross product is that its resulting vector is orthogonal (perpendicular) to both of the original vectors. This ensures that the cross product effectively captures a direction that is normal to the plane defined by the two vectors.

Vector Cross Product

The cross product is a binary operation on two vectors in three-dimensional space that results in a new vector. This new vector is perpendicular to the plane containing the original vectors, and its magnitude is proportional to the area of the parallelogram spanned by the original vectors.

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