If vectors 𝐀 and 𝐁 are given by 𝐀=5 𝐢̂+6 𝐣̂+3 𝐤 and 𝐁=6 𝐢̂-2 𝐣̂-6 𝐤̂. Which is/are of the follow

step 1 So the question says if factors A and B are given by A equals 55 plus 6J plus 3K and B equals 6i

If vectors 𝐀 and 𝐁 are given by 𝐀=5 𝐢̂+6 𝐣̂+3 𝐤 and 𝐁=6 𝐢̂-2...

If vectors $\mathbf{A}$ and $\mathbf{B}$ are given by $\mathbf{A}=5 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}+3 \mathbf{k}$ and $\mathbf{B}=6 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$. Which is/are of the following correct? (a) A and B are mutually perpendicular (b) Product of $A \times B$ is the same $B \times A$ (c) The magnitude of $\mathrm{A}$ and $\mathrm{B}$ are equal (d) The magnitude of $A \cdot B$ is zero

If vectors $\mathbf{A}$ and $\mathbf{B}$ are given by $\mathbf{A}=5 \hat{\mathbf{i}}+6 \hat{\mathbf{j}}+3 \mathbf{k}$ and $\mathbf{B}=6 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$. Which is/are of the following correct?
(a) A and B are mutually perpendicular
(b) Product of $A \times B$ is the same $B \times A$
(c) The magnitude of $\mathrm{A}$ and $\mathrm{B}$ are equal
(d) The magnitude of $A \cdot B$ is zero

Key Concepts

Orthogonality

Orthogonality in vector spaces refers to the concept of perpendicularity between vectors. Two vectors are orthogonal if their dot product is zero, signifying that they have no component in common in any direction. This concept is widely used in various fields, including geometry, physics, and computer graphics, to simplify analyses and solve problems involving independent directions.

Vector Magnitude

The magnitude of a vector, also known as its norm or length, is a measure of the distance of the vector from the origin in a vector space. Calculated as the square root of the sum of the squared components, vector magnitude is used to quantify distances, normalize vectors, and compare different vectors in terms of size.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It tightly connects to the concept of the angle between vectors; when the dot product of two vectors equals zero, the vectors are orthogonal or perpendicular. This concept is fundamental in determining directional relationships in vector spaces.

Cross Product

The cross product is a binary operation on two vectors in three-dimensional space and produces a vector that is perpendicular to both of the original vectors. A key property of the cross product is its anti-commutativity, meaning that switching the order of the vectors reverses the sign of the resulting vector. This operation is essential in applications involving torque, rotational dynamics, and finding normals to surfaces.

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