Let’s dive into these topics one by one:
Position and Displacement Vectors
- Position Vector: This represents the location of a point relative to an origin in a coordinate system. For example, if you have a point PPP with coordinates (x,y)(x, y)(x,y) in a 2D plane, the position vector r\mathbf{r}r of PPP can be written as r=xi^+yj^\mathbf{r} = x\hat{i} + y\hat{j}r=xi^+yj^, where i^\hat{i}i^ and j^\hat{j}j^ are the unit vectors in the x and y directions, respectively. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
- Displacement Vector: This represents the change in position of a point from an initial position to a final position. If point AAA has position vector rA\mathbf{r}_ArA and point BBB has position vector rB\mathbf{r}_BrB, the displacement vector d\mathbf{d}d from AAA to BBB is given by d=rB−rA\mathbf{d} = \mathbf{r}_B – \mathbf{r}_Ad=rB−rA. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
General Vectors and Their Notations
- Vector Notation: Vectors are usually denoted by bold letters (e.g., v\mathbf{v}v) or letters with an arrow on top (e.g., v⃗\vec{v}v). For example, a vector v\mathbf{v}v in 2D can be expressed as v=vxi^+vyj^\mathbf{v} = v_x \hat{i} + v_y \hat{j}v=vxi^+vyj^, where vxv_xvx and vyv_yvy are the vector’s components along the x and y axes. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
Equality of Vectors
- Equality: Two vectors are equal if they have the same magnitude and direction. Mathematically, if v=a\mathbf{v} = \mathbf{a}v=a and w=b\mathbf{w} = \mathbf{b}w=b, then v=w\mathbf{v} = \mathbf{w}v=w if and only if a=b\mathbf{a} = \mathbf{b}a=b. In component form, this means vx=axv_x = a_xvx=ax and vy=ayv_y = a_yvy=ay (in 2D) or similarly for 3D vectors. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
Multiplication of Vectors by a Real Number
- Scalar Multiplication: When a vector v\mathbf{v}v is multiplied by a scalar kkk, the result is a new vector kvk \mathbf{v}kv whose magnitude is scaled by ∣k∣|k|∣k∣ and whose direction is either the same (if kkk is positive) or opposite (if kkk is negative). For instance, if v=vxi^+vyj^\mathbf{v} = v_x \hat{i} + v_y \hat{j}v=vxi^+vyj^ and kkk is a scalar, then kv=(kvx)i^+(kvy)j^k \mathbf{v} = (k v_x) \hat{i} + (k v_y) \hat{j}kv=(kvx)i^+(kvy)j^. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
Addition and Subtraction of Vectors
- Addition: To add two vectors v\mathbf{v}v and w\mathbf{w}w, you simply add their corresponding components. For example, if v=vxi^+vyj^\mathbf{v} = v_x \hat{i} + v_y \hat{j}v=vxi^+vyj^ and w=wxi^+wyj^\mathbf{w} = w_x \hat{i} + w_y \hat{j}w=wxi^+wyj^, then v+w=(vx+wx)i^+(vy+wy)j^\mathbf{v} + \mathbf{w} = (v_x + w_x) \hat{i} + (v_y + w_y) \hat{j}v+w=(vx+wx)i^+(vy+wy)j^. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
- Subtraction: To subtract one vector w\mathbf{w}w from another vector v\mathbf{v}v, subtract their corresponding components. For instance, v−w=(vx−wx)i^+(vy−wy)j^\mathbf{v} – \mathbf{w} = (v_x – w_x) \hat{i} + (v_y – w_y) \hat{j}v−w=(vx−wx)i^+(vy−wy)j^. Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
Let me know if you’d like to go over specific examples or any other details! Momentum Coaching is the top 5 best IIT-JEE, NEET Coaching in Varanasi.
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