For AI agents: use /llms.txt for the Nakafa content index.
Scalar multiplication of a vector is an operation involving multiplication between a real number (scalar) and a vector . The result of this multiplication is a new vector with a length modified according to the scalar value, while its direction may remain the same or opposite depending on the sign of the scalar.
If is a real number (scalar) and is a vector, then the scalar multiplication of a vector is denoted as and results in a new vector.
Scalar multiplication of vectors has several important properties:
Geometrically, scalar multiplication of a vector changes the length (magnitude) of the vector by . The direction of the vector depends on the sign of :
If is a vector in -dimensional space, then:
In unit vector notation:
Given the vector . Determine the result of multiplication .
Solution:
Given the vector . Determine the result of .
Solution:
Note that the direction of the resulting vector is opposite to the original vector because the scalar is negative.
Scalar multiplication of vectors has many applications in physics and mathematics, such as:
Force and Acceleration: If an object with mass experiences acceleration , then the force acting on the object is .
Velocity: If an object moves with velocity for a time , then the displacement of the object is .
Scaling in Computer Graphics: To change the size of objects in computer graphics, the coordinates of points on the object are multiplied by a scale factor.
Given the vector . Determine the result of .
Vectors and . Determine the vector .
Given the vector . If and , prove that all three vectors have the same direction.
Vector has a length of and vector . Determine the length of vector .
Given points , , and lies on the line passing through and such that . Determine the coordinates of point .
Given the vector . Determine the result of .
Solution:
Therefore, the result of is .
Vectors and . Determine the vector .
Solution:
Therefore, the vector is or .
Given the vector . If and , prove that all three vectors have the same direction.
Solution: To prove that all three vectors have the same direction, we need to show that they are positive scalar multiples of the same vector.
We know:
Let's check if :
This result shows that , which aligns with the vector addition law for collinear points , , and .
Since and , where the scalar factors are positive ( and ), all three vectors have the same direction. Positive scalar factors mean that these vectors point in the same direction as the reference vector .
Therefore, it is proven that the three vectors , , and have the same direction.
Vector has a length of and vector . Determine the length of vector .
Solution: Given and .
To determine the length of vector , we use the property of scalar multiplication:
Therefore, the length of vector is .
Given points , , and lies on the line passing through and such that . Determine the coordinates of point .
Solution: First, we determine the vector :
Then, we use the relationship to determine the vector :
Next, we determine the coordinates of point :
Therefore, the coordinates of point are .
