Скачать или смотреть PROBLEM 12 BASED ON CONDITION OF PERPENDICULARITY OF TWO VECTORS

Basic Concepts: - Scalar: A quantity with only magnitude (e.g., mass, length).
Vector: A quantity that has both magnitude and direction (e.g., displacement, velocity).
Position Vector: A vector that represents the position of a point relative to the origin, typically denoted as r⃗ = xi+yj+zk
Zero Vector: A vector with zero magnitude and no specific direction.
2. Vector Representation: - A vector a⃗ in three-dimensional space can be written as: a⃗=a1i+a2j+a3k, where i^, j^, k^ are the unit vectors along the x, y, and z axes, respectively.
3. Magnitude of a Vector: - The magnitude or length of a vector a⃗=a1i^+a2j^+a3k^ is given by: ∣a⃗∣ = sqrt{a1^2 + a2^2 + a3^2}
4. Unit Vector: - A unit vector has a magnitude of 1 and is in the direction of the given vector. The unit vector a^ in the direction of a⃗ is:a^ = a⃗/∣a⃗∣
Addition and Subtraction of Vectors: - Addition: a⃗+b⃗=(a1+b1)i^+(a2+b2)j^+(a3+b3)k^
Subtraction: a⃗−b⃗=(a1−b1)i^+(a2−b2)j^+(a3−b3)k^
5. Scalar (Dot) Product: - The scalar product of two vectors a⃗ and b⃗ is: a⃗⋅b⃗=a1b1+a2b2+a3b3=∣a⃗∣∣b⃗∣cosθ, where θ is the angle between the vectors.
If a⃗⋅b⃗=0, then a⃗ and b⃗ are perpendicular (orthogonal).
6. Vector (Cross) Product: - The vector product of two vectors a⃗ and b⃗ is: a⃗×b⃗=(a2b3−a3b2)i^+(a3b1−a1b3)j^+(a1b2−a2b1)k^
The magnitude of the cross product is: ∣a⃗×b⃗∣=∣a⃗∣∣b⃗∣sinθ
where θ is the angle between a⃗ and b⃗, and the direction is given by the right-hand rule.
7. Triple Product: - Scalar Triple Product: For three vectors a⃗, b⃗, c⃗ the scalar triple product is: a⃗⋅(b⃗×c⃗) = V This gives the volume of the parallelepiped formed by the vectors a⃗,b⃗,c⃗.
8. Vector Triple Product: For three vectors, the vector triple product is: a⃗×(b⃗×c⃗)=(a⃗⋅c⃗)b⃗−(a⃗⋅b⃗)c⃗
9. Projection of a Vector: - The projection of vector a⃗ onto vector b⃗ is: Projb⃗a⃗=a⃗⋅b⃗ b⃗|b⃗^2|
10. Collinearity of Vectors: - Two vectors a⃗ and b⃗ are collinear if a⃗ = kb⃗ for some scalar k.
11. Coplanarity of Vectors: - Three vectors a⃗, b⃗, c⃗ are coplanar if their scalar triple product is zero: a⃗⋅(b⃗×c⃗)=0.
12. Position Vector of a Point Dividing a Line Segment: - If a point P divides a line segment joining two points A and B in the ratio m : n, the position vector of P is: OP⃗=mB⃗+nA⃗/m+n

Types of Vectors: -
1. Zero Vector (Null Vector): - A vector with zero magnitude and no specific direction. Represented as 0⃗. Its components are all zero: 0⃗=0i^+0j^+0k^. Used to denote the origin or when two vectors cancel each other out.
2. Unit Vector: - A vector with a magnitude of 1. It indicates only direction, without concern for magnitude. If a⃗ is a vector, its unit vector is given by a^ =a⃗/∣a⃗∣. Examples of unit vectors are i^, j^, k^ which are unit vectors along the x, y, and z axes, respectively.
3. Position Vector: - A vector that represents the position of a point relative to the origin. The position vector of a point P(x,y,z) in space is given by OP⃗=xi^ + yj^ + zk^
4. Collinear Vectors: - Vectors that are parallel to the same line or have the same or opposite direction. If vectors a⃗ and b⃗ are collinear, then a⃗ = kb⃗ for some scalar k.
5. Equal Vectors: - Two vectors are said to be equal if they have the same magnitude and direction, regardless of their initial points. If a⃗ and b⃗ are equal, a⃗ = b⃗.
6. Negative of a Vector: - A vector that has the same magnitude as the given vector but opposite direction. If a⃗ is a vector, its negative is denoted by -a⃗ and a⃗+(−a⃗)=0⃗.
7. Parallel Vectors: - Vectors that have the same or exactly opposite directions. They may have different magnitudes but will always be scalar multiples of each other.
8. Orthogonal Vectors: - Vectors that are perpendicular to each other. Two vectors a⃗ and b⃗ are orthogonal if their dot product is zero: a⃗⋅b⃗=0.
9. Coplanar Vectors: - Three or more vectors that lie in the same plane. For three vectors a⃗, b⃗, c⃗ to be coplanar, their scalar triple product must be zero: a⃗⋅(b⃗×c⃗) = 0.
10. Displacement Vector: - vector that represents the change in position of a point from one point to another. If a point moves from position P(x1,y1,z1) to Q(x2,y2,z2), the displacement vector is given by: PQ→=(x2−x1)i^+(y2−y1)j^+(z2−z1)k^
11. Reciprocal Vector: - A vector that is directed along the same line as the given vector but whose magnitude is the reciprocal of the given vector's magnitude. If a⃗ is a vector, its reciprocal vector b⃗ is given by b⃗ = a^/∣a⃗∣.
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