Math Problem Statement

IV. Duoti vektoriai \(\vec{a} = (2; -1; 3)\) ir \(\vec{b} = -\vec{i} - 2\vec{j} + \vec{k}\). Apskaičiuokite: \((\vec{a} + 2\vec{b})^2\).

Solution

The problem provided involves vectors and asks to compute (a+2b)2(\vec{a} + 2\vec{b})^2, where the vectors a=(2,1,3)\vec{a} = (2, -1, 3) and b=i2j+k\vec{b} = -\vec{i} - 2\vec{j} + \vec{k} are given.

Solution:

  1. Write the vectors:

    • a=(2,1,3)\vec{a} = (2, -1, 3)
    • b=(1,2,1)\vec{b} = (-1, -2, 1) (from i2j+k-\vec{i} - 2\vec{j} + \vec{k}).
  2. Compute 2b2\vec{b}: 2b=2(1,2,1)=(2,4,2).2\vec{b} = 2(-1, -2, 1) = (-2, -4, 2).

  3. Add a+2b\vec{a} + 2\vec{b}: a+2b=(2,1,3)+(2,4,2)=(0,5,5).\vec{a} + 2\vec{b} = (2, -1, 3) + (-2, -4, 2) = (0, -5, 5).

  4. Square the resulting vector ((a+2b)2(\vec{a} + 2\vec{b})^2): The square of a vector is the dot product of the vector with itself: (a+2b)2=(0,5,5)(0,5,5).(\vec{a} + 2\vec{b})^2 = (0, -5, 5) \cdot (0, -5, 5).

  5. Compute the dot product: (a+2b)2=02+(5)2+52=0+25+25=50.(\vec{a} + 2\vec{b})^2 = 0^2 + (-5)^2 + 5^2 = 0 + 25 + 25 = 50.

Final Answer:

(a+2b)2=50.(\vec{a} + 2\vec{b})^2 = 50.

Would you like further details or step-by-step explanations? Here are some related questions for practice:

  1. How is the dot product of two vectors defined and computed?
  2. Can you calculate (ab)2(\vec{a} - \vec{b})^2 for the same vectors a\vec{a} and b\vec{b}?
  3. What is the magnitude of the vector a+2b\vec{a} + 2\vec{b}?
  4. What is the geometric meaning of squaring a vector?
  5. How would the calculation change if b\vec{b} had a different coefficient, such as 3b3\vec{b}?

Tip: Always simplify vectors step-by-step to avoid errors in addition or scalar multiplication.

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Math Problem Analysis

Mathematical Concepts

Vector Algebra
Dot Product
Vector Operations

Formulas

Vector addition: \(\vec{u} + \vec{v} = (u_x + v_x, u_y + v_y, u_z + v_z)\)
Scalar multiplication: \(c\vec{u} = (cu_x, cu_y, cu_z)\)
Dot product: \(\vec{u} \cdot \vec{v} = u_xv_x + u_yv_y + u_zv_z\)

Theorems

Properties of the dot product
Distributive property of vector addition

Suitable Grade Level

Grades 10-12