Question

Find \(\|\vec{u}\|,\|\vec{v}\|,\|\vec{u}+\vec{v}\|\) and \(\|\vec{u}-\vec{v}\|\) \(\vec{u}=\langle 2,-3,6\rangle, \quad \vec{v}=\langle 10,-15,30\rangle\)

Short answer

\(\|\vec{u}\| = 7\), \(\|\vec{v}\| = 35\), \(\|\vec{u} + \vec{v}\| = 42\), \(\|\vec{u} - \vec{v}\| = 28\).

Step-by-step solution

Calculate the Magnitude of \( \vec{u} \)

To find \( \|\vec{u}\| \), use the formula \( \|\vec{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2} \). For \( \vec{u} = \langle 2, -3, 6 \rangle \), the calculation is \( \|\vec{u}\| = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \).

Calculate the Magnitude of \( \vec{v} \)

To find \( \|\vec{v}\| \), use the formula \( \|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \). For \( \vec{v} = \langle 10, -15, 30 \rangle \), the calculation is \( \|\vec{v}\| = \sqrt{10^2 + (-15)^2 + 30^2} = \sqrt{100 + 225 + 900} = \sqrt{1225} = 35 \).

Calculate \( \vec{u} + \vec{v} \)

To find \( \vec{u} + \vec{v} \), add corresponding components: \( \langle 2, -3, 6 \rangle + \langle 10, -15, 30 \rangle = \langle 2+10, -3-15, 6+30 \rangle = \langle 12, -18, 36 \rangle \).

Calculate the Magnitude of \( \vec{u} + \vec{v} \)

For \( \|\vec{u} + \vec{v}\| \), use the formula: \( \|\vec{u} + \vec{v}\| = \sqrt{(u_1+v_1)^2 + (u_2+v_2)^2 + (u_3+v_3)^2} \). With \( \langle 12, -18, 36 \rangle \), compute \( \|\vec{u} + \vec{v}\| = \sqrt{12^2 + (-18)^2 + 36^2} = \sqrt{144 + 324 + 1296} = \sqrt{1764} = 42 \).

Calculate \( \vec{u} - \vec{v} \)

To calculate \( \vec{u} - \vec{v} \), subtract corresponding components: \( \langle 2, -3, 6 \rangle - \langle 10, -15, 30 \rangle = \langle 2-10, -3+15, 6-30 \rangle = \langle -8, 12, -24 \rangle \).

Calculate the Magnitude of \( \vec{u} - \vec{v} \)

For \( \|\vec{u} - \vec{v}\| \), use the formula: \( \|\vec{u} - \vec{v}\| = \sqrt{(u_1-v_1)^2 + (u_2-v_2)^2 + (u_3-v_3)^2} \). With \( \langle -8, 12, -24 \rangle \), compute \( \|\vec{u} - \vec{v}\| = \sqrt{(-8)^2 + 12^2 + (-24)^2} = \sqrt{64 + 144 + 576} = \sqrt{784} = 28 \).

Key concepts

Vector Addition

Vectors are entities that possess both magnitude and direction. Vector addition involves taking two vectors and summing their respective components to create a new vector. This process is essential in fields like physics for calculating forces or velocities that are acting together. To add vectors, you simply add each of the corresponding components.
Suppose we have vectors \( \vec{u} = \langle 2, -3, 6 \rangle \) and \( \vec{v} = \langle 10, -15, 30 \rangle \). To find \( \vec{u} + \vec{v} \), we add each component of \( \vec{u} \) to the corresponding component of \( \vec{v} \):

• First component: \( 2 + 10 = 12 \)
• Second component: \( -3 - 15 = -18 \)
• Third component: \( 6 + 30 = 36 \)

This results in the vector \( \vec{u} + \vec{v} = \langle 12, -18, 36 \rangle \).
Vector addition is commutative, meaning \( \vec{u} + \vec{v} = \vec{v} + \vec{u} \), and associative, \( (\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w}) \).
This ensures consistent results, regardless of how the vectors are grouped or ordered in computation.

Vector Subtraction

Just like vector addition, vector subtraction is a crucial operation in linear algebra which involves subtracting the corresponding components of one vector from another. It is often used to find the difference between two vectors or to reverse direction in applications like navigation and physics.
For the vectors \( \vec{u} = \langle 2, -3, 6 \rangle \) and \( \vec{v} = \langle 10, -15, 30 \rangle \), the subtraction \( \vec{u} - \vec{v} \) is calculated as follows:

• First component: \( 2 - 10 = -8 \)
• Second component: \( -3 + 15 = 12 \)
• Third component: \( 6 - 30 = -24 \)

Therefore, the resulting vector is \( \vec{u} - \vec{v} = \langle -8, 12, -24 \rangle \).
Unlike addition, vector subtraction is not commutative, meaning \( \vec{u} - \vec{v} eq \vec{v} - \vec{u} \).
It follows certain properties such as \( \vec{u} - \vec{v} = \vec{u} + (-\vec{v}) \), where \(-\vec{v}\) involves negating every component of \( \vec{v} \).

Magnitude Calculation

The magnitude of a vector is a measure of its length or size in space, without regard to its direction. It is particularly helpful in physics and engineering to determine the strength or intensity of quantities like forces and velocities.
To calculate the magnitude of a 3-dimensional vector \( \vec{a} = \langle a_1, a_2, a_3 \rangle \), use the formula:
\[ \|\vec{a}\| = \sqrt{a_1^2 + a_2^2 + a_3^2} \]
For instance, the magnitude of \( \vec{u} = \langle 2, -3, 6 \rangle \) is:

• Calculate each squared component: \( 2^2 = 4 \), \( (-3)^2 = 9 \), \( 6^2 = 36 \)
• Sum the squares: \( 4 + 9 + 36 = 49 \)
• Take the square root: \( \sqrt{49} = 7 \)

Thus, \( \|\vec{u}\| = 7 \).
Similarly, for \( \vec{v} = \langle 10, -15, 30 \rangle \), its magnitude is calculated as \( \|\vec{v}\| = 35 \).
Understanding magnitude is key to comparing vectors and can be extended to more complex vector operations.

Euclidean Norm

The Euclidean norm, often just called "the norm" or sometimes referred to as the \( L2 \) norm, is a specific way of computing the size of a vector, similar to the magnitude. It's the most common norm used in both mathematics and physics for its straightforward calculation and foundational role in vector spaces.
The Euclidean norm for a vector \( \vec{a} = \langle a_1, a_2, \, \ldots, \ a_n \rangle \) in \( n \)-dimensional space is:
\[ \|\vec{a}\| = \sqrt{a_1^2 + a_2^2 + \ldots + a_n^2} \]
This represents a generalization of the Pythagorean theorem, describing the distance of the vector from the origin to the point \( (a_1, a_2, \ldots, a_n) \) in Euclidean space.
When applied to our example vector \( \vec{u} = \langle 2, -3, 6 \rangle \), here’s a quick rundown:

• Square each component: \( 2^2 = 4 \), \( (-3)^2 = 9 \), \( 6^2 = 36 \)
• Add the squares: \( 4 + 9 + 36 = 49 \)
• Square root the sum: \( \sqrt{49} = 7 \)

The Euclidean norm is synonymous with the magnitude for vectors and gives insight into their spatial extent and directionality.