If A⃗-Ax î+Av ĵ+Az k⃗, B⃗-Bx î+By ĵ+B k̇ we have Column-1 Column-II (a) A⃗ ·B⃗ (p) Ax=Bx^', Ay=B

step 1 Hello everyone, we have a vector given as A vector. It can be written as AX i cap plus AY J cap

If A⃗-Ax î+Av ĵ+Az k⃗, B⃗-Bx î+By ĵ+B k̇ we have Column-1...

If $\vec{A}-A_{x} \hat{i}+A_{v} \hat{j}+A_{z} \vec{k}, \vec{B}-B_{x} \hat{i}+B_{y} \hat{j}+B \dot{k}$ we have Column-1 Column-II (a) $\vec{A} \cdot \vec{B}$ (p) $A_{x}=B_{x^{\prime}}, A_{y}=B_{y} A_{z}=B_{z}$ (b) $\vec{A}-\vec{B}$ (q) $\vec{A} \cdot \vec{B}=0$ (c) $\vec{A} \uparrow \uparrow \vec{B}$ (r) $A_{\pi} B_{x}+A_{y} B_{r}+A_{z} B_{z}$ (d) $\vec{A} \perp \vec{B}$ (s) $\sqrt{\left(A_{x}^{2}\left|A_{y}^{2}\right| A_{z}^{2}\right)\left(B_{x}^{2}\left|B_{y}^{2}\right| B_{z}^{2}\right)}$

If $\vec{A}-A_{x} \hat{i}+A_{v} \hat{j}+A_{z} \vec{k}, \vec{B}-B_{x} \hat{i}+B_{y} \hat{j}+B \dot{k}$ we have
Column-1 Column-II
(a) $\vec{A} \cdot \vec{B}$
(p) $A_{x}=B_{x^{\prime}}, A_{y}=B_{y} A_{z}=B_{z}$
(b) $\vec{A}-\vec{B}$
(q) $\vec{A} \cdot \vec{B}=0$
(c) $\vec{A} \uparrow \uparrow \vec{B}$
(r) $A_{\pi} B_{x}+A_{y} B_{r}+A_{z} B_{z}$
(d) $\vec{A} \perp \vec{B}$
(s) $\sqrt{\left(A_{x}^{2}\left|A_{y}^{2}\right| A_{z}^{2}\right)\left(B_{x}^{2}\left|B_{y}^{2}\right| B_{z}^{2}\right)}$

Key Concepts

Vector 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. In terms of components, it is calculated by multiplying corresponding components and then summing those products. The result is a scalar and the dot product also has a geometric interpretation as the product of the magnitudes of the two vectors and the cosine of the angle between them.

Vector Subtraction

Vector subtraction is the operation of subtracting one vector from another by subtracting corresponding components. This operation results in a new vector, which represents the difference between the original vectors, and is useful for determining relative positions or differences in force, velocity, and other vector quantities.

Parallel Vectors

Vectors are said to be parallel if they have the same or exactly opposite direction. Mathematically, two vectors are parallel when one is a scalar multiple of the other. In terms of components, the ratios of the corresponding components are equal, which is a necessary condition for parallelism.

Perpendicular (Orthogonal) Vectors

Two vectors are perpendicular, or orthogonal, if the angle between them is 90 degrees. This condition is characterized mathematically by the dot product being zero. Orthogonality is a key concept in various areas of mathematics and physics, as it indicates independence in the directions of the vectors.

Magnitude (Norm) of a Vector

The magnitude or norm of a vector is a measure of its length and is calculated as the square root of the sum of the squares of its components. This concept is fundamental in vector analysis and is used to describe the size or extent of a vector in space.

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