Open Question
Find a vector that points in the same direction as the vector ( î ＋ ĵ ) and whose magnitude is 1.
3. Vectors
Unit Vectors
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A = (4.0 m)i + (3.0 m)j and B = (−13.0 m)i + (7.0 m)j. You add them together to produce another vector C.
(a) Express this new vector C in unit-vector notation.
(b) What are the magnitude and direction of C?
A
C = (9 m) i + (10 m) j = 13.5 m @ 48° above the +x-axis
B
C = -(9 m) i + (10 m) j = 13.5 m @ 48° above the -x-axis
C
C = (9 m) i + (10 m) j = 13.5 m @ 0.83° above the +x-axis
D
C = -(9 m) i + (10 m) j = 13.5 m @ 0.83° above the -x-axis
Verified step by step guidance1
To find the vector C, add vectors A and B component-wise. For the i-component, add the i-components of A and B: (4.0 m) + (-13.0 m). For the j-component, add the j-components of A and B: (3.0 m) + (7.0 m).
Express the resulting vector C in unit-vector notation using the results from the previous step. C = (i-component) i + (j-component) j.
To find the magnitude of vector C, use the Pythagorean theorem: magnitude = sqrt((i-component)^2 + (j-component)^2).
To find the direction of vector C, calculate the angle θ with respect to the positive x-axis using the tangent function: θ = arctan((j-component)/(i-component)).
Determine the correct quadrant for the angle based on the signs of the i and j components, and adjust the angle θ accordingly to express the direction of vector C.
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